ph.d thesis model

��)Ä:�

ÿÀÆùÂ

º d

uÀ��ÆêÆX

�"�ocÊ�

ccc óóó�ùÂ´±ù�Vd�5ÿÀÆÄ:�§6�7�¤>��ù�v. �Oy´ùÇ

:8ÿÀÚ�êÿÀÐÚ�SN. Ï��, 8c��¹:8ÿÀ�SN, ¿��

�U�\0�:8ÿÀ¥��(Ø.

�ö�~a�Æ)®I[�n�ùÂ�tex��, Ùó�þ´�~ã��. Ó�,

�ö�a�ÐlY�u�·Jøù��Å¬ùÇ (:8) ÿÀÆ.

8 ¹

8 ¹

1�Ù �Ö: ÿÀÆ{0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 �o´ÿÀÆº . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 ÿÀÆ�{¤u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 ÿÀÆ�©a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1�Ù :8ÿÀ (I): ÿÀ�m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 ÿÀ�m�m8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 ÿÀ�m��E�{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 �{�: ÿÀÄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 �{�: SÿÀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.3 �{n: ÈÿÀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.4 �{o: f�mÿÀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.5 �{Ê: ÝþÿÀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15�ÙSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1nÙ :8ÿÀ (II): ÿÀ�Ä�5� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 4��à: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Hausdorff 5� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 ëÏ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 ;�5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 4�:;�S�; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 ëYN� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.1 ëYN��Ó� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.2 ëYN��E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6.3 ëYN��ëÏ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.4 ëYN��;5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6.5 ëYN��Ýþ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

1oÙ :8ÿÀ (III): �\E| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.1 �ê5ún . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 ©l5ún . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Urysohn Ún� Tietze *Ü½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Urysohn Ýþz½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Tychonoff ½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ë�©z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

- ii -

1�Ù �Ö: ÿÀÆ{0

1�Ù �Ö: ÿÀÆ{0

1.1 �o´ÿÀÆº

3·��ªùÇù��c, k{ü0��eÿÀÆ´�o. ��ÿÀÆ��½5�)º, ·�ÄkÚ\ÿÀC��Vg. oÑ/`, §´�ã/�� !� (�¦Øe�§ØÊÜ)C�. ¤¢ÿÀÆ (Topology), Ò´ïÄã/3ÿÀC�e�±ØC�5� (�¡��AÛÆ).ùp^B`K{. Ù¢·�ÆL�éõAÛÆÑ�±w�´ïÄAÛã/3,�aAÛC

�e�±ØC�5�. ù´�«��AÛg�.

AÛC� �±ØC�5� éAAÛÆ

fNC� �Ýþ (�Ý!�Ý!¡È) î¼AÛ��C� ��'X� ��AÛ©ª�5C� ��,��±,��'� �üAÛ(E²¡)

�KC� ��'� �KAÛÿÀC� ��ê,�ëÏ5� ÿÀÆ��IC� �±½�� ©AÛ

·�UY£�ÿÀÆ�{Kþ. ÿÀÆ�ª48I´�òã/3Ó�¿Âe©a. ùp¤

¢�Ó�, ´�ü�ã/�ÏLÿÀC�*dpC. ¢yù�8I´�~(J�¯. ¢Sþ�k3�ê��¹eâU)û©a¯K. ·�¬3�¡î�½ÂÓ��Vg.

1.2 ÿÀÆ�{¤u

ùp~Þ�ÿÀÆ�u ¯K.

~ 1.2.1 (�)x¯K) ²¡þdº:Ú>�¤�ã(Graph )UØUd�)x¤ (=�¦

ØE�Ø¢¦/rH¤k�>Úº:)? ù�¯K�@dî.)û, ´ãØÚÿÀÆ�²;u ¯K��. é��ã5`, �)x¯K¿Ø�6uã¥�>´��½ö�, Ï ´��ÿÀ

¯K. �

~ 1.2.2 (àõ¡N½n) ��àõ¡N�º:ê� E, cê� F , ¡ê V . î.äóX

eð�ªE − F + V = 2.

ù�´@ÏÿÀÆ�²;(Ø��. ��e, XJ·�rõ¡N��¡�K¿^å.mù��f, r��õ¡NØA�S¡þ, @oõ¡NÒC¤²¡þ�ã. Ïdî.½n��±w¤'uã�ÿÀ½n. �

~ 1.2.3 (oÚ¯K) �/ãþ��«�XÚ,�¦��I[kØÓ�ôÚ. ¯��I�A

«ôÚ÷v±þ�¦? ù�¯K��Y´: �I� 4 «ôÚÒv. ù�¯KÄk�8(�ãØ¯K, ,�dO�Å��y�a�/. �

- 1 -

1�Ù �Ö: ÿÀÆ{0

~ 1.2.4 (#'¿d�) ò�^Ý/��^�àÛ= 180 Ý, �Ùé>ÊÜ, ��¡�#'¿d�. §Ú�ÎkX��ØÓ�AÛ (ÿÀ) 5�. 'X, §´üý¡. �´�Î%´Vý¡.

,, ��#'¿d�þÕk��< (Þ�þ), ¦l,�:Ñu, ÷X#'¿d�r��£��:. @o\¬uy¦�ÞC��e��. ù3êÆþ��Ø�½�5. ù«5�¢Sþ�N#'¿d�Ú�Î�m�ÿÀ(��É. �ö�k²T�ÿÀ(�, cö%k�²T�ÿÀ(�. �¡·�¬?�Ú�\&?§. �

ÿÀÆ�ý�CÄ<´�êÆ[\4. ¦mM|ÜÿÀÆ, �ÑÍ¶�\4éó½n, ¿�Ú\Ä�+�Vg-ÿÀÆ��êÆé��-��.

1.3 ÿÀÆ�©a

UìDÚ�©a, ÿÀÆ��±©�o�©|: :8ÿÀ!�êÿÀ!|ÜÿÀ!�©ÿÀ. :8ÿÀ5gu¢ê8ÚëY¼ê�5� ('X0�½n�). �êÿÀ�¹ÓNØÜÓÔ

Ô, Ù¥ÓNØ5 uî.àõ¡N½n, ÓÔØK5 u\4'uÄ�+�ïÄ. |ÜÿÀ

¢Sþ�±w¤�êÿÀ��Ü©, 5 u|ÜÓNØ. �©ÿÀKïÄÛÜ�©5�Ú�NÿÀ�m�'X, 'XÍ¶�pd-ÆBAúª.

ÿÀÆ®²�~2�/'ß��êÆ©|p. 'X, ØAÛÆ�, §��/A^�ÃX�¼©Û!VÇÚO!¢C¼ê!�©�§��nØ¥.

- 2 -

1�Ù :8ÿÀ (I): ÿÀ�m

1�Ù :8ÿÀ (I): ÿÀ�m

2.1 ÿÀ�m�m8

·�Äk£�êÆ©Û¥¢ê¶ X = R1 þm8�Vg.

(1) X þ�m«m´�Xe/ª�8Ü

U1 = (a, b)4= {x ∈ X | a < x < b}.

AO/, ·��±ò�8��5½ ∅ = (1, 0).

(2) m��:

U2 :=(a,+∞)4= {x ∈ R1 | x > a},

U3 :=(−∞, b)4= {x ∈ R1 | x < b}.

�8�U�¤ X = (−∞,+∞).

(3) ��m8½Â��m«m�¿8. 'X, (−1, 0) ∪ (2, 3) ´m8. ¢Sþm��Ú�8

�U�¤m«m�¿.

(a,+∞) =⋃

n ∈ Z+

n > a

(a, n), (−∞, b) =⋃

n ∈ Z+

n > −b

(−n, b), X =⋃

n∈Z+

(−n, n).

¢ê¶þ�m8÷v±en^5�:

(1) X, ∅ ´m8,

(2) ?¿õ�m8�¿E´m8,

(3) k�õ�m8��E´m8.

5 2.1.1 5� (3) ¥“k�õ�” �^�ØU�, 'X:⋂n∈Z+

(− 1n,1n

) = {0}

Ø´m8.

y3, ·��l¢ê¶m8�VgÑu, ½ÂÄ��ÿÀ�mÚm8�Vg.

½Â 2.1.1 � X ´��8Ü, T ´ X þ�f8�¤�8x, ÷v±e^�:

(1) ∅ ∈ T , X ∈ T ,

(2) T ¥?¿õ��¿�3 T ¥,

(3) T ¥k�õ��3 T ¥,

- 3 -

1�Ù :8ÿÀ (I): ÿÀ�m

K¡ T ´ X þ��ÿÀ (Topology), X ¡�ÿÀ�m. T ¥��¡�m8 (Open set).

e¡Þ�ÿÀ�m�~f.

~ 2.1.1 (¢ê¶þ�IOÿÀ) � X = R1, T = {U | U ´m«m�¿8}. w, T ´8Ü X �ÿÀ, T ¥��=�Ï~n)�m8. ù�ÿÀ¡�IOÿÀ. �

~ 2.1.2 (²¡þ�IOÿÀ) � X = R2, T = {U | U ´m��¿8}, T �´ X �IOÿÀ, Ùm8�·�3êÆ©Û¥n)�Vg��. �

~ 2.1.3 � X = {1, 2, 3}. ·��±½Â X þ�«ØÓ�ÿÀ.

(1) T1 = {∅, X}. ù´²��ÿÀ,

(2) T2 = { ∅, X, {1}, {2}},

(3) T3 = X ��8 ( =¤kf8�¤�x),

(4) T4 = {∅, X, {2}, {1, 2}, {2, 3}}. �

5 2.1.2 (1) þ~L² X þ�UkNõØÓ�ÿÀ.

(2) ¿�?Û8xÑ´ÿÀ. 'X X = {1, 2, 3} þ

T = { ∅, X, {1, 2}, {2, 3}}

¿�ÿÀ. ù´Ï� {2} = {1, 2} ∩ {2, 3} Ø3 T ¥. �

k�8Üþ�ÿÀkNõk��|ÜêÆ¯K. 'X

¯K 2.1.1 � Xn = {1, 2, · · · , n}, @o Xn þkõ�«ØÓ�ÿÀ?

~ 2.1.4 (lÑÿÀ) � X ´��8Ü, T ´ X ��8. TÿÀ¡�lÑÿÀ.

~ 2.1.5 (²�ÿÀ) � X ´��8Ü, T = {∅, X} ½Â�ÿÀ¡�²�ÿÀ.

~ 2.1.6 ({k�ÿÀ) � X ´Ã�8Ü,

Tf = {U | �o U = ∅,�o X − U ´k�8}.

·�5�y§´ÿÀ.

(1) d½Â: ∅ ∈ Tf . Ï� X −X = ∅, ��´k�8, ¤± X ∈ Tf .

(2) � {Uα}α∈I ⊆ Tf , ·��y⋃

α∈I

Uα ∈ Tf , =y X −⋃

α∈I

Uα´k�8. du

X −⋃α∈I

Uα =⋂α∈I

(X − Uα),

¿� X − Uα ´k�8, � X −⋃

α∈I

Uα ´k�8, l X −⋃

α∈I

Uα ∈ Tf

- 4 -

1�Ù :8ÿÀ (I): ÿÀ�m

(3) � U1, U2, . . . Un ∈ Tf (= X − Ui ´k�8). d

X −n⋂

i=1

Ui =n⋃

i=1

(X − Ui)

í�n⋃

i=1(X − Ui) ´k�8. Ïd

n⋂i=1

Ui ∈ Tf .

nþ¤ã, Tf ´ X þ�ÿÀ. �

aq/, ·��½ÂXeÿÀ�m.

~ 2.1.7 ({�êÿÀ) � X´Ø�ê8Ü,

Tf = {U | �o U = ∅,�o X − ∅ ´�ê8}.

�ÖögC�yù´ÿÀ�m. �

½Â 2.1.2 � X ��, T1 Ú T2 ´ X þ�ü�ÿÀ. e T1 ⊆ T2, K¡ T2 [u T1, ½¡T1 ou T2.

~ 2.1.8 ²�ÿÀoulÑÿÀ.

~ 2.1.9 � X = {1, 2, 3},

T1 = {∅, X, {1}, {1, 2}}, T2 = {∅, X, {1}, {2}, {1, 2 }, {2, 3}},

K T1 ⊆ T2, Ïd T1 ou T2.

2.2 48

½Â 2.2.1 � T ´ X �ÿÀ, Y ⊆ X, e X − Y ∈ T ´m8, K¡ Y ´48 (closeset).

~ 2.2.1 � X = R1, T ´IOÿÀ. ·��4«m [a, b] := {x | a ≤ x ≤ b}. Ï�

X − [a, b] = (−∞, a) ∪ (b,+∞) ´ T �m8, ¤± [a, b] ´48.

~ 2.2.2 T ´ X þ�lÑÿÀ,é?Ûf8 Y ⊆ X, Y ´m8. ,��¡, X − Y ∈ T ,Ïd Y´48. nþ, YQ´m8,q´48.

~ 2.2.3 � X = {1, 2, 3}, T = {∅, X, {1}, {2, 3}}, Y = {1} ´m8. ,��¡, X − Y ={2, 3} ∈ T L² Y ´48. Ïd YQ´m8�´48.

~ 2.2.4 X = R1, Tf ´{k�ÿÀ.Y ⊆ X, Y ´48��=� X − Y ´m8, = X − (X − Y ) ´k�8, ½ö X − Y = ∅, ½= Y

´k�8½ö Y = X.

- 5 -

1�Ù :8ÿÀ (I): ÿÀ�m

~ 2.2.5 X = R2, T ´ X þ�IOÿÀ. �

Y = {(x, y) | x ≥ 0, y ≥ 0}.

Ï� X − Y = (−∞, 0)×R1 ∪R1 × (−∞, 0) ´m8, ¤± Y ´48.

·K 2.2.1 X ´��ÿÀ�m, K

(1) ∅, X ´48,

(2) ?¿õ�48��´48,

(3) k�õ�48�¿´48.

y² (1) Ï� X − ∅ = X ∈ T , � ∅ ´48. qÏ X −X = ∅ ∈ T , ¤± X �´48.

(2) � {Yα}α∈I ´�x48, Uα = X − Yα. d½Â, Uα ∈ T ´m8. 5¿

X −⋂α∈I

Yα =⋃α∈I

Uα

´m8, �⋂

α∈I

Yα ´48.

(3) � Y1, Y2, . . . , Yn ´48. du

X − Y1 ∪ Y2 ∪ · · · ∪ Yn =n⋂

i=1

Ui

´n⋂

i=1Ui ´m8, � Y1 ∪ Y2 ∪ · · · ∪ Yn ´48. �

5 2.2.1 �8Ü X ´��8, ·��±^“48” ½Â X þ�ÿÀ. äN�{Xe: �C ´f8x,÷v:

(1) X, ∅ ∈ C ,

(2) C ¥?¿õ��8E3 C ¥,

(3) C ¥k�õ��¿8�E3 C ¥.

- T = {U | X − U ∈ C }, K T �Ñ8Ü X þ�ÿÀ.

~ 2.2.6 (Zariski ÿÀ) � X = Cn ´Eê�þ n ��m. �Äõ�ª�§|:

f1(x1, x2, . . . , xn) = 0f2(x1, x2, . . . , xn) = 0· · · · · · · · · · · · · · · · · · · · · · · ·fn(x1, x2, . . . , xn) = 0

½ÂT�§|�)8� Z(f1, f2, · · · , fr). w,k

Z(f1, f2, . . . , fr) = Z(f1) ∩ Z(f2) ∩ · · · ∩ Z(fr).

- 6 -

1�Ù :8ÿÀ (I): ÿÀ�m

·�P U(f1, f2, . . . , fr) = X − Z(f1, f2, . . . , fr),

T = {¤kùa U(f1, f2, . . . , fr)}, C = {¤kõ�ª�§|)8}.

±e·�äó T ´ÿÀ, ¡�� Zariski ÿÀ. §´�êAÛ¥�Ä��ïÄé�.

|^5P 2.2.1 9·K 2.2.1, ·��I��y C ´48x, l §p� X þ�ÿÀ T .

Äk5¿� ∅ = Z(1) ( =�§ 1 = 0 Ã)) 9X = Z(0), Ïd ∅, X ∈ C .

- {Yα}α∈I ⊆ C . d½Â��

Yα = Z(fα1 , fα2 , . . . , fαrα) = Z(fα1) ∩ Z(fα2) ∩ · · · ∩ Z(fαrα

)

Ïd ⋂α∈I

Yα =⋂α∈I

(Z(fα1) ∩ Z(fα2) ∩ · · · ∩ Z(fαrα)) = Z({fαβ

})

d²;�(Ø, õ�ª� C[x1, · · · , xn] ¥dÃ {fαi} )¤�n��±^k��)¤. �ó

�, �§| {fαi= 0} ¥�±]Ñk��§, §��)8Ú {fαi

= 0} �)8��. Ïd⋂α∈I

Yα ∈ C .

� C ¥k�� Y1, Y2, . . . , Yn ∈ C . 8yn⋃

i=1Yi ´��48. dêÆ8B{, ·��I

y² n = 2 ��/. Ø��5, �

Y1 = Z(f1, f2, . . . , fr), Y2 = Z(g1, g2, . . . , gl)

@o

Y1 ∪ Y2 = Z

{fi · gj} 1 ≤ i ≤ r ,

1 ≤ j ≤ l

∈ C . (2-1)

nþ,·�y² T ´ X þ�ÿÀ. �

~ 2.2.7 � X = C, T1 ´ Zariski ÿÀ, T2 ´{k�ÿÀ. dpd�êÆÄ�½n, ·�k T1 = T2. �ÖögC�y. �

2.3 ÿÀ�m��E�{

2.3.1 �{�: ÿÀÄ

½Â 2.3.1 X ´��8Ü, B ´ X �f8x, ÷v±e^�

(1) ?� x ∈ X, �3 U ∈ B ¦� x ∈ U ,

(2) � x ∈ U1 ∩ U2, ùp U1, U2 ∈ B, K�3 U3 ∈ B ¦� x ∈ U3 ⊆ U1 ∩ U2.

·�¡ B ´ X ��ÿÀÄ. ÿÀÄ B ¥��¡�Ä��.

- 7 -

1�Ù :8ÿÀ (I): ÿÀ�m

|^ÿÀÄ, ·��±�EÑÿÀ. ù��E�{k:aqu^�5Ã'�þ|�E�þ�m.

½Â 2.3.2 � B ´ X �ÿÀÄ, T ´ X �f8x, ÷v:

U ∈ T ⇐⇒ U = ∅ ½ U ´ B ¥Ä��¿,

K T ¡�d B )¤�ÿÀ.

�ÖögC�yþã� T (¢´ÿÀ. ^ÿÀÄ£ãÿÀw,��Béõ. ±e·��~f.

~ 2.3.1 � X = R1, T ´IOÿÀ, B = {¤k�m«m}. ÿÀÄ B )¤T . �

~ 2.3.2 � X = R2, T ´IOÿÀ, B = {¤k�m��}, ÿÀÄ B )¤ T . �

·K 2.3.1 � B ´ÿÀ�m X �ÿÀÄ. T ´��f8x, @o±eü�^�*d�

d:

(1) T ´ B )¤�ÿÀ,

(2) ?� U ∈ T , é?¿ x ∈ U , �3 B ∈ B ¦� x ∈ B ⊆ U .

�L5§éu�½�ÿÀ�m§XÛ�ä��m8x´Ä´ù�ÿÀ�Ä? e¡�(Ø£�ù�¯K.

·K 2.3.2 � X ´ÿÀ�m, T ´ÿÀ. � B ´ X �m8x, ÷v±e^�: é?Ûm8

U 9 x ∈ U , �3 B ∈ B, ¦� x ∈ B ⊆ U . @o B ´TÿÀ�Ä.

y² é?Û x ∈ X, du X �´m8, �db�^��, �3 B ∈ B, ÷v x ∈ B ⊆ X.

� B1, B2 ∈ B, x ∈ B1 ∩ B2. Ï� B1, B2 ´m8, ¤± B1 ∩ B2 �´m8. db�^�, �

3 B3 ∈ B, ÷v x ∈ B3 ⊆ B1 ∩B2. �

|^ÿÀÄ'�ü�ÿÀ�o[�¬�Béõ.

·K 2.3.3 � X ´��ÿÀ�m, T ,T ′ ´ X þ�ÿÀ, B,B′ ©O´ T ,T ′ �ÿÀÄ,K±en�^�*d�d:

(1) T ′[u T ( = T ⊆ T ′),

(2) é?¿ x ∈ X 9�¹ x �?¿Ä�� B ∈ B,Ñ�3 B′ ∈ B′ ÷v x ∈ B′ ⊆ B,

(3) B ¥?ÛÄ��Ñ´ B′ ¥Ä��¿.

y² (2) � (3) ��d5´w,�, Ïd�Iy² (1) � (2) ��d5.

(1) ⇒ (2) ?� x ∈ X 9�¹ x �Ä�� B ∈ B. Ï�

B ⊆ T ⊆ T ′,

¤± B ∈ T ′. d·K 2.3.1 9b�^�, �3 B′ ∈ B′ ¦� x ∈ B′ ⊆ B.

(2) ⇒ (1) du B ´ T �ÿÀÄ, ��Iy² B ⊆ T ′ =�. ∀B ∈ B, db�^� (3), B´ B′ ¥Ä��¿, = B ∈ T ′. �

- 8 -

1�Ù :8ÿÀ (I): ÿÀ�m

~ 2.3.3 � X = R2, B1 = {¤k�m��}. §´ÿÀÄ, )¤IOÿÀ T1.

� B2 = {¤k�mÝ/}, §�)¤ÿÀP� T2. |^·K 2.3.3, N´�y T1 = T2. �

~ 2.3.4 (e�ÿÀ) � X = R1, B1 = {¤k�m«m}, §)¤IOÿÀ T1.

� T2 = {¤k��m«m [a, b)}, §)¤¤¢�e�ÿÀ T2.

·��y T1 $ T2. ?� (a, b) ∈ B1 9?¿ x ∈ (a, b), �3 B2 ¥�Ä�� [x, b) ⊆ (a, b),¦� x ∈ [x, b). d·K 2.3.3, T1 ou T2. ��, �Ä B2 ¥�Ä�� [c, d) 9- x := c ∈ [c, d).d�Ø�3 B1 ¥�?ÛÄ�� (a, b) ¦� c ∈ (a, b) ⊆ [c, d). Ïd T1 6= T2. �

2.3.2 �{�: SÿÀ

½Â 2.3.3 � X ´��8Ü. e38Ü X þ�3��S'X <, ÷v±e^�:

(1) (�'�5) ∀ x, y ∈ X, x 6= y, K�o x < y �o y < x,

(2) (�g�5) ∀x ∈ X, x < x Ø�U¤á.

(3) (D45) ∀x, y, z ∈ X, ek x < y 9 y < z, K x < z ¤á,

K X �¡��S8.

~ 2.3.5 � X = R1,

(1) X þ�~^S'X <: x < y ⇔ y − x ∈ R+.

(2) ½Â X þ�,�S'X: x < y ⇔ �o |x| < |y| �o |x| = |y| � x < y ( �ÖögC�y

ù´��S'X). �

~ 2.3.6 (i;S'X) ®� (X,<X) 9 (Y,<Y ) ´ü��S8, ·�½Â X Ú Y �(k�¦È

Z = X × Y = {(x, y) | x ∈ X, y ∈ Y }.

3 Z þ½Â�S'X <Z : (x1, y1) <Z (x2, y2) ⇔ �o x1 <X x2, �o x1 = x2 � y1 <Y y2. ù�S'X¡�i;S'X.

aq/, ·��±½Âõ��S8�(k�Èþ�i;S'X. �

½Â 2.3.4 («m) � (X,<X) ´'u <X ��S8, a, b ∈ X, a < b.

(1) m«m

(a, b) def= {x ∈ X | a < x < b}.

AO/, e (a, b) = ∅, K¡ a ´ b �;�c�, b ´ a �;��.

(2) 4«m

[a, b] def= {x ∈ X | a ≤ x ≤ b}.

- 9 -

1�Ù :8ÿÀ (I): ÿÀ�m

(3) �m«m

[a, b) def={x ∈ X | a ≤ x < b},

(a, b] def={x ∈ X | a < x ≤ b}.

½Â 2.3.5 (�� (�) �) � (X,<X) ´�S8.

(1) e�3 a ∈ X ¦�é?¿ x ∈ X Ñk a ≤ x, K¡ a ´��.

(2) e�3 b ∈ X ¦�é?¿ x ∈ X Ñk x ≤ b, K¡ b ´��.

kùVg�, ·��±�EÿÀÄ.

½Â 2.3.6 (SÿÀ) � B ´ÿÀ�m X þ��f8x. U ∈ B ��=� U ´±ea.�«m��:

(1) U = (a, b);

(2) U = [a0, b) (e�� a0 �3);

(3) U = (a, b0] (e�� b0 �3).

B ´ÿÀÄ, )¤�ÿÀ¡�SÿÀ.

�e5·��yþã B (¢´ÿÀÄ.

y² Äk�y, é?Û x ∈ X, Ñ�3�¹ x ��Ä��. ±e©�/?Ø:

(1) x Ø´�� (�) �. d��é a, b ∈ X ¦� a < x < b, = x ∈ (a, b).

(2) x ´��, x ∈ [x, b), ùp b ∈ X ´?�÷v b > x ��.

(3) x ´��, x ∈ (a, x],ùp a ∈ X ´?� a < x ��.

Ùg, é?¿ U1 = (a, b), U2 = (c, d) ∈ B, N´�y U1 ∩ U2 E´ B ¥��. �

·K 2.3.4 þã B )¤ÿÀ T ( ¡�SÿÀ ).

~ 2.3.7 X = R1, ~^S'X½Â�ÿÀÄ B )¤ R1 þ�IOÿÀ. �

~ 2.3.8 (i;SÿÀ) X = R1 × R1 þdi;S½Â�ÿÀÄP� (a × b, c × d), ùpx× y L«�I (±�Ú��þ«m�PÒ· ). �

~ 2.3.9 X = Z+, < ´kd~^S'X½Â��S8. 1 ´��. 5¿�ü:8

{n} =

{(n− 1, n+ 1), n > 1,

[1, 2), n = 1.

Ïdz�ü:8 {n} Ñ´Ä��, l Z+ þ�SÿÀ´Ò´lÑÿÀ. �

- 10 -

1�Ù :8ÿÀ (I): ÿÀ�m

~ 2.3.10 � X = {0, 1} × Z+, ·�^ Pn L«�� 0× n, ^ Qn L«�� 1× n. u´

X = {P1, P2, · · · , Pn · · · , Q1, Q2, · · · , Qn, · · · } .

�Äþ¡�i;S <. ·�k

Pn < Qn, Pn < Pm, Qn < Qm, n < m.

ü:8

{Pn} =

{(Pn−1, Pn+1) n > 1,[P1, P2) n = 1

´m8. aq/, {Qn} (n > 1) �´m8. �´ {Q1} Ø´m8, Ï��¹ Q1 �?Ûm«m7¹k,� Pi. ÏdþãSÿÀØ´��lÑÿÀ. �

½Â 2.3.7 � X ´�S8, ·�½Â

(1) m��

(a,+∞) def= {x | a < x} =⋃x>a

(a, x),

(−∞, a) def= {x | x < a} =⋃x<a

(x, a)

(2) 4��

[a,+∞) def= {x | a ≤ x} =⋃x≥a

(a, x],

(−∞, a] def= {x | x ≤ a} =⋃x≤a

[x, a)

5 2.3.1 (1) m��w,´SÿÀ¥�m8.(2) e a0 ´��, @o(−∞, a) = [a0, a). aq/, e a0 ´��, @o(a,+∞) = (a, a0]. �

2.3.3 �{n: ÈÿÀ

� X,Y ´ÿÀ�m, ½Â X � Y �(k�È

Z = X × Y4= {(x, y) | x ∈ X, y ∈ Y }.

�E Z þ�ÿÀ, ·��I��EéA�ÿÀÄ=�. � U ⊆ X (�A/, V ⊆ Y ) ´ X (�A/, Y ) ¥�m8. ·��E Z þ�f8 U × V . y3�ÄXe8x

B = {W ⊆ Z |W = U × V,ùp U, V ©O´ X,Y ¥�m8}.

·�5y²þã� B ´ Z þ�ÿÀÄ, l p� Z þ�ÿÀ, ¡��ÈÿÀ(ProductTopology).

Äk, ?� (x, y) ∈ Z. d X,Y ��ÿÀ, �3 X (�A/, Y ) ¥�m8 U ⊆ X (�A/,V ⊆ Y ) ÷v x ∈ U (�A/, y ∈ V ). Ïd (x, y) ∈ U × V . Ùg, ·��

B1 = U1 × V1, B2 = U2 × V2 ∈ B,

- 11 -

1�Ù :8ÿÀ (I): ÿÀ�m

�Ø�� (x, y) ∈ B1 ∩B2.-

B3 = (U1 ∩ U2)× (V1 ∩ V2) ∈ B.

·�k (x, y) ∈ B1 ∩B2 = B3. nÜ��, B ´ Z ��|ÿÀÄ.

~ 2.3.11 � X = Y = R1 Ñ´�kIOÿÀ�¢ê8, Z = X × Y = R2 þ�ÈÿÀÒ´²¡þ�IOÿÀ. �

d·K 2.3.2, N´�yXe(Ø.

·K 2.3.5 b� B1 ´ X ��|ÿÀÄ, B2 ´ Y ��|ÿÀÄ, Z = X × Y , K

B3 = {W ⊆ Z |W = B1 ×B2, B1 ∈ B1, B2 ∈ B2}

´ÈÿÀ Z = X × Y ��ÿÀÄ.

~ 2.3.12 (ÝKN�) ·�kg,�ÝKN�

X × Ypr1−→ X, X × Y

pr2−→ Y,

(x, y) 7−→ x, (x, y) 7−→ y.

� U ⊆ X, V ⊆ Y ©O´ X,Y ¥�m8, @o

pr−11 (U) ={(x, y) | x ∈ U, y ∈ Y } = U × Y,

pr−12 (V ) ={(x, y) | x ∈ X, y ∈ V } = X × V

w,´ Z ¥�m8, ¿�÷v pr−11 (U) ∩ pr−1

2 (V ) = U × V . �

~ 2.3.13 (�ÿÀ) � {Xα}α∈I ´�xÿÀ�m,

Z =∏α∈I

Xα4= {(xα)α∈I | xα ∈ Xα},

B = {W ⊆ Z |W =∏α∈I

Uα, Uα ´ Xα ¥�m8}

aqþ¡?Ø, B �´��ÿÀÄ, )¤ Z þ�ÿÀ, ·�¡��ÿÀ(Box topology).

d��ÝKN�P�

prα : Z −→ Xα, (xα)α∈I 7−→ xα.

XJ {Xα}α∈I ´dk��ÿÀ�m�¤�, @o·��r∏α∈I

Xα ¡�ÈÿÀ. �

Öö�U¬¯, �Û·�Ø��òþãÿÀ¡�“ÈÿÀ”Q? ¢Sþ, 3��/�(k�Èþ, ·�UXe�ª½ÂÈÿÀ.

½Â 2.3.8 (2ÂÈÿÀ) �Ä Z =∏α∈I

Xα þ�f8x

B = {W ⊆ Z |W =∏α∈I

Uα, ùp Uα ´ Xα �m8, ¿�Øk�� α , Ñk Uα = Xα}.

B ´ Z þ�ÿÀÄ (�ÖögC�y), )¤�ÿÀ¡� Z �ÈÿÀ.

- 12 -

1�Ù :8ÿÀ (I): ÿÀ�m

3k�(k�È�/, �ÿÀÚÈÿÀ��, vk7�«©. 3��/, �ÿÀ�[uÈÿÀ. 3ÿÀÆ�ïÄ¥, ÈÿÀ�5��Ð. k�È�/�éõ�(ØÃ{í2��/��ÿÀþ, �%�±í2�ÈÿÀþ. Ïd·�ÀJò�ö¡�ÈÿÀ��Ü·.

2.3.4 �{o: f�mÿÀ

½Â 2.3.9 � (X,T ) ´ÿÀ�m, Y ⊆ X ´��f8,

TY4= {Y ∩ U | U ⊆ X´m8}.

TY ¡� Y þ�f�mÿÀ(Subspace topology).

k��Lã�B, 3Ø�u· ��¹e, ·��òf�mÿÀ�¤ T |Y , {¡� T 3 Y þ��.

e¡·�5y² TY (¢�Ñ Y þ��ÿÀ.

y² (1) d

∅ =∅ ∩ Y ∈ TY ,

Y =X ∩ Y ∈ TY

á� ∅ ∈ TY , Y ∈ TY .

(2) � {Uα ∩ Y }α∈I ⊆ TY . Ï� Uα ´ X �m8, ¤±⋃

α∈I

Uα �´ X �m8, �⋃α∈I

(Uα ∩ Y ) = Y ∩ (⋃α∈I

Uα) ∈ TY .

(3) �U1 ∩ Y, U2 ∩ Y, · · · , Un ∩ Y

´ TY ¥��. Ï�n⋂

k=1

Uk ´m8, ¤±

n⋂k=1

(Uk ∩ Y ) = (n⋂

k=1

Uk) ∩ Y ∈ TY .

nÜ±þ, TY ´ Y �ÿÀ, =�f�mÿÀ. �

·K 2.3.6 � B ´ X �ÿÀÄ, BY = {B ∩ Y | B ∈ B}, K BY ´ Y þ�f�mÿÀ�Ä.

e¡·�|^·K 2.3.2 5�yþã(Ø.

y² � U ´ X ¥�?�m8, Y ∩ U ∈ TY . Ø�� Y ∩ U ��.

?� y ∈ Y ∩ U , ·�é� B ∈ B, ¦� y ∈ B ∩ Y ⊆ Y ∩ U =�. dÿÀÄ�½Â, ·�w

,�±é� B ∈ B ¦� y ∈ B ⊆ U . §÷vþã�¦. d·K 2.3.2 =�¤I(Ø. �

��`5, Y ¥�m8�7´ X ¥�m8. 'Xe¡�{ü~f.

- 13 -

1�Ù :8ÿÀ (I): ÿÀ�m

~ 2.3.14 �Ä X = R1 þ�IOÿÀ. �

Y = [0, 1], U = (12,32),

@o Y ∩ U = (12 , 1] ´ Y þf�mÿÀ¥�m8, �¿Ø´ X ¥�m8.

e¡·�5��e Y �f�mÿÀ�Ä.

(a, b) ∩ Y =

(a, b), e (a, b) ⊆ Y,

[0,b), e a < 0 < b ≤ 1,(a,1], e 0 ≤ a < 1 < b,

Y, e Y ⊆ (a, b),∅, Ù¦.

ù�~fL², d? Y �SÿÀ¢SþÚ§�f�mÿÀ´��. �

~ 2.3.15 �Ä X = R1 T þ�IOÿÀ. � Y = [0, 1) ∪ {2}. 3 Y �f�mÿÀ TY ¥,ü:8

{2} = (32,52) ∩ Y

´ Y ¥�m8.

25� Y þ�SÿÀ T ′. � B ´ T ′ ¥¹k 2 �Ä��

(a, 2]T ′ = {y ∈ Y | a < y ≤ 2} = (a, 2]X ∩ Y.

�âSÿÀÄ��5½, a ∈ Y , ¿� a < 2, Ï 0 < a < 1. ù�, þã«m��¹ Y ¥,�Ø�u 2 ��. Ïd {2} Ø´ Y �SÿÀ¥�m8.

ù�~fL², Y �f�mÿÀ�7Ú§g��SÿÀ��. �

~ 2.3.14 �(Ø3�½^�e�±í2��SÿÀþ.

·K 2.3.7 (SÿÀ��) (X,T ) ´SÿÀ�m, Y ⊆ X ´SÿÀe�m«m(½m�

�), @of�mÿÀ TY � Y þ�SÿÀ��.

(�ÖögC�y)

~ 2.3.16 (X,T ) ´lÑÿÀ, Y ⊆ X, @of�mÿÀ TY Ò´ Y �lÑÿÀ. �

·K 2.3.8 (ÈÿÀ��) � X,Y ´ÿÀ�m, A,B ©O´ X Ú Y ¥�f8. � T ´X × Y þ�ÈÿÀ, T ′ ´ A × B þ�ÈÿÀ (A,B ©Oäk X,Y �f�mÿÀ). @o A × B

�f�mÿÀ� T ′ ��. �ó�, ·�kXe'Xª@o·�k±e'X¤á:

TA×B = T ′.

5 2.3.2 ��B�[PÁ, ·��±òþã(Ø{��

TA×B = TA ×TB,

ùp TA (�A/, TB) L« A (�A/, B) ��ÿÀ�m X (�A/, Y ) �f�mÿÀ. �

e¡·�{�/�yù�(Ø.

- 14 -

1�Ù :8ÿÀ (I): ÿÀ�m

y² Ø�� U × V ´ X × Y �ÿÀÄ¥�Ä��, d½Â,

(U × V ) ∩ (A×B) ∈ TA×B.

,��¡,(U × V ) ∩ (A×B) = (U ∩A)× (V ∩B) ∈ T ′.

Ïd TA×B ⊆ T ′.

Ón�U�� T ′ ⊆ TA×B. Ïdü�ÿÀ´��. �

2.3.5 �{Ê: ÝþÿÀ

ù�!ò0��«�EÿÀ�²;�ª. §´ÏL¯k�½�Ýþ5p�ÑÿÀ. ùaÿÀ

é�CuêÆ©Û¥�~�ÿÀ. �éÙ¦ÿÀ5`, §��5��´L.

Äk£��eÝþ�Vg.

½Â 2.3.10 8Ü X þ�Ýþ(Metric)

d : X ×X −→ R1

´�÷v±e^��¼ê:

(1) (�½5)d(x, y) ≥ 0, ∀ x, y ∈ X,

¿� d(x, y) = 0 ��=� x = y.

(2) (é¡5)d(x, y) = d(y, x), ∀x, y ∈ X.

(3) (n�Ø�ª)d(x, y) + d(y, z) ≥ d(x, z), ∀x, y, z ∈ X.

·�¡ d(x, y) ´ x, y 'uÝþ d �ål (Distance).

d, é?¿ x ∈ X ±9?¿�¢ê ε, ·�½Â± x �¥%� ε-¥

Bd(x; ε)4= {y ∈ X | d(x, y) < ε}.

�Ä8xB = {¤k� ε-¥}.

·�òäóXe(Ø.

·K 2.3.9 B ´ X þ�ÿÀÄ. §)¤�ÿÀ¡�dÝþ d p��ÝþÿÀ (Metrictopology).

y² (1) ∀x ∈ X, � Bd(x, 1), w, x ∈ Bd(x, 1)

(2) � B1 = Bd(x1, ε1), B2 = Bd(x2, ε2), �b� B1 ∩B2 6= ∅.

- 15 -

1�Ù :8ÿÀ (I): ÿÀ�m

éu ∀x ∈ B1 ∩B2, ·�F"é�� ε-¥ B3 = Bd(x, δ) ¦�

x ∈ B3 ⊆ B1 ∩B2.

·�� δ1 = ε− d(x1, x). é?¿ z ∈ Bd(x, δ1), d½Â� d(x, z) < δ1. |^n�Ø�ª��

d(x1, z) ≤ d(x, x1) + d(x, z) < d(x1, x) + δ1 = ε1.

Ï Bd(x, δ1) ⊆ B1. Ón, �é�¥ Bd(x, δ2) ⊆ B2.

8� δ = min(δ1, δ2). - B3 = Bd(x, δ). u´

x ∈ Bd(x, δ) ⊆ Bd(x1, δ1) ∩Bd(x, δ2) ⊆ B1 ∩B2

÷v¤I^�. �

~ 2.3.17 X = R1 þkIOÝþ d(x, y) = |x− y|.

Bd(x, ε) = (x− ε, x+ ε) = {y ∈ R1 | |y − x| < ε}.

d��ÝþÿÀÒ´ R1 þ�IOÿÀ. �

~ 2.3.18 X = R2 þ�IOÝþ d(x,y) = ||x− y|| Ò´�Ï~�î¼Ýþ. ε-¥

Bd(x, ε) = {y ∈ R2 | ||y − x|| < ε}

Ò´m��. d��ÝþÿÀÒ´ R2 þ�IOÿÀ. �

~ 2.3.19 � X ´��8Ü, ·�½ÂÝþ (�ÖögCy²)

d(x, y) =

{1 x 6= y

0 x = y

ù�Ýþp�lÑÿÀ. ¯¢þ, é?Û x ∈ X,

Bd

(x,

12

)= {x}.

Ï z�ü:8Ñ´m8. �

íØ 2.3.1 � (X, d) ´Ýþ�m, Bd(x, ε) ´ ε-¥. é?¿: y ∈ Bd(x, ε), o�3¥Bd(y, δ) ÷v

y ∈ Bd(y, δ) ⊆ Bd(x, ε).

y² - y ´ B1 = Bd(x, ε) ¥?�:, B2 = Bd(y, 1). w�, y ∈ B1 ∩B2.

d·K 2.3.9 �y², �é�¥ B3 = Bd(y, δ) ¦�

y ∈ Bd(y, δ) ⊆ B1 ∩B2 ⊆ B1.

ùÒ�¤y². �

~ 2.3.20 ·�£� X = Rn ¥�î¼Ýþ. �Äü�:��I

x = (x1, x2, · · · , Xn), y = (y1, y2, · · · , yn).

IO�î¼Ýþ½Â�

d(x,y)4=

√(x1 − y1)2 + (x2 − y2)2 + · · ·+ (xn − yn)2.

- 16 -

1�Ù :8ÿÀ (I): ÿÀ�m

k��P� ||x− y||. �

�!��òy²Xe�(Ø.

½n 2.3.1 (î¼�m��Ýþz) Rn þ�ÈÿÀ� Rn þ�ÝþÿÀ�Ó.

5 2.3.3 lþ¡�½n, <��±JÑ��k��¯K: ��ÿÀ�m X þ´Äo�3Ýþ d, ¦� d p��ÝþÿÀTÐÒ´ X �ÿÀQ? þã¯K��Y´Ä½�. XJ X þ�

3ù��Ýþ, ·�Ò` X ´�Ýþz�, ¡��Ýþ�m. ��ÿÀ�mÛ�´�Ýþz�?ù´��¯K. �©ò�0��Ýþz½n�Ñ£�. �

3y²½n 2.3.1 �c, ·�k��O�ó�.

½Â 2.3.11 (�») � (X, d) ´Ýþ�m, A ⊆ X ´��f8. ·�½Â A ��»

d(A) = sup{d(a, b) | a, b ∈ A}.

e d(A) <∞, K¡ A ´k.�.

A �k.5r��6uÝþ�À�, Ïdù¿Ø´ÿÀ5�. ·�ò3e©y², ?ÛÝþÑ�±^,�k.ÝþO�, §�äk�Ó�ÿÀ.

·�kQãXe(Ø. §3'�ÝþÿÀ��, �~¢^.

Ún 2.3.1 (ÝþÿÀ'��K) � d, d′ ´ X þ�ü«Ýþ, T ,T ′ ©O´§�p��Ý

þÿÀ, @o±e^��d:

(1) T ⊆ T ′

(2) é?¿ x ∈ X 9?¿ ε > 0, o�3 δ > 0, ¦� Bd′(x, δ) ⊆ Bd(x, ε).

y² (1)=⇒(2) � T ¥� ε-¥ Bd(x, ε). Ï� T ⊆ T ′ � Bd(x, ε) ∈ T , ¤± Bd(x, ε) ∈T ′. Ï �3 Bd′(x, δ) ∈ T ′ ¦� x ∈ Bd′(x, δ) ⊆ Bd(x, ε).

(2)=⇒(1) ?� T ¥ ε-¥ Bd(y, r) ±9?�: x ∈ Bd(y, r). díØ 2.3.1, �3 Bd(x, ε) ÷

vx ∈ Bd(x, ε) ⊆ Bd(y, r).

db�^�, ·��é�¥ Bd′(x, δ), ¦�

x ∈ Bd′(x, δ) ⊆ Bd(x, ε) ⊆ Bd(y, r).

d·K 2.3.3, ùÒíÑ T ⊆ T ′. �

·K 2.3.10 (k.Ýþ) � (X, d) ´Ýþ�m, d(x, y) = min{d(x, y), 1}, K

(1) d �´Ýþ;

(2) d Ú d p�Ñ�Ó�ÿÀ.

- 17 -

1�Ù :8ÿÀ (I): ÿÀ�m

y² (1) d ��½5�é¡5´w,�, ·�5y²n�Ø�ª

d(x, y) ≤ d(x, z) + d(y, z).

e d(x, z), d(y, z) ¥��u�u 1, 'X d(x, z) ≥ 1, @o

d(x, y) ≤ 1 ≤ 1 + d(y, z) = d(x, z) + d(y, z).

e d(x, z) < 1 � d(y, z) < 1, K d(x, z) = d(x, z), d(y, z) = d(y, z), K

d(x, y) ≤ d(x, y) ≤ d(x, z) + d(y, z) = d(x, z) + d(y, z).

nþ=�, d ´Ýþ.

(2) �âÚn 2.3.1, ·��yXe'X:

Bd(x, ε) ⊆ Bd(x, ε), (2-2)

Bd(x, δ) ⊆ Bd(x, ε), (2-3)

Ù¥ δ = min{ε, 1}.

kyª (2-2). � z ∈ Bd(x, ε), = d(x, z) < ε. Ï k

d(x, z) ≤ d(x, z) < ε.

ùÒíÑ z ∈ Bd(x, ε). Ïd Bd(x, ε) ⊆ Bd(x, ε).

2yª (2-3). � z ∈ Bd(x, δ), ùp δ = min{1, ε}, Ï d(x, z) < δ ≤ 1. ù%¹X

d(x, z) = d(x, z) < δ ≤ ε.

Ïd z ∈ Bd(x, ε). dd�� Bd(x, δ) ⊆ Bd(x, ε).

nþ, d Ú d p�Ñ�Ó�ÿÀ. �

~ 2.3.21 � X = Rn,

x = (x1, x2, · · · , xn), y = (y1, y2, · · · , yn).

d(x,y) EL«î¼Ýþ.

·�½Â X þ�²�Ýþ

ρ(x,y) := max{|x1 − y1|, |x2 − y2|, · · · , |xn − yn|}.

(1) �y: ρ ´��Ýþ.

·��I�yn�Ø�ª. � z = (z1, z2, · · · , zn). Ï�

|xi − zi| ≤ |xi − yi|+ |yi − zi| ≤ ρ(x,y) + ρ(y,z),

¤±

ρ(x,z) = maxi{|xi − zi|} ≤ ρ(x,y) + ρ(y,z).

(2) �y

ρ(x,y) ≤ d(x,y) ≤√nρ(x,y).

- 18 -

1�Ù :8ÿÀ (I): ÿÀ�m

Ï�

d(x,y) =

√√√√ n∑i=1

(xi − yi)2 ≥ |xi − yi|

é¤k i ¤á, ��1��Ø�ª.

,�Ø�ª5gu

d(x,y) =

√√√√ n∑i=1

(xi − yi)2 ≤√n ·max

i{(xi − yi)2} =

√n ∗ ρ(x,y).

(3) �y Bd(x, ε) ⊆ Bρ(x, ε).

é?¿ y ∈ Bd(x, ε), Ï� d(x,y) < ε, �d (2) �

ρ(x,y) ≤ d(x,y) < ε,

= y ∈ Bρ(x, ε). ù�Ò��¤I(Ø.

(4) �y Bρ(x, ε√n) ⊆ Bd(x, ε)

� z ∈ Bρ(x, ε√n), = ρ(x,z) < ε√

n. d (2) �,

d(x,z) ≤√n · ρ(x,z) < ε,

= z ∈ Bd(x, ε). dd=�(Ø.

(5) d (3) Ú (4) ±9ÝþÿÀ�'��K, ρ � d p��Ó�ÿÀ. �

½n2.3.1 �y²: |^þ¡�?Ø, ·��I�y² X �ÈÿÀ (P� Tp) �²�Ýþ ρ

p��ÝþÿÀ (P� Tρ) �Ó.

�B = (a1, b1)× · · · × (an, bn)

´ÈÿÀ�Ä��, x = (x1, · · · , xn) ∈ B. éz� i, ·�� εi > 0, ÷v

(xi − εi, xi + εi) ⊆ (ai, bi).

- ε = min{ε1, · · · , εn}, Kx ∈ Bρ(x, ε) ⊆ B.

Ïd Tp ⊆ Tρ.

�L5, é?¿�¥ Bρ(x, ε) ±9?�: y ∈ Bρ(x, ε), ρ(x,y) < ε ��u |xi − yi| < ε éz� i ¤á. 8� Tp �Ä��

B = (x1 − ε1, x1 + ε1)× · · · (xn − εn, xn + εn).

ù�, y ∈ B ⊆ Bρ(x, ε). Ïd Tρ ⊆ Tp. �

�ÙSK\ ∗ Ò�SKL«k�½JÝ.

SK 2.1

- 19 -

1nÙ :8ÿÀ (II): ÿÀ�Ä�5�


3.1 4��à:

·K 3.1.1 X ´ÿÀ�m, Y ´ X ��8Ü, A ⊆ Y , K±e^�*d�d:

(1) A ´ Y ¥�48;

(2) �3 X ¥�48 C, ¦� A = Y ∩ C.

5 3.1.1 þã·K�^� (1) ´�3 Y �f�mÿÀ¿Âe. ,, ^� (2) ¥�48 C

¿Ø��. �

y² (1)=⇒(2) � B = Y −A ´ Y ¥�m8, df�mÿÀ�½Â, �3 X ¥�m8 U ,¦� B = Y ∩ U . u´

A = Y −B = Y − (Y ∩ U) = Y ∩ (X − U)

Ï� X − U ´ X ¥�48, ¤±�� C = X − U , (Ø¤á.

(2)=⇒(1) ®��3 X ¥�48 C ¦� A = Y ∩ C. - B = Y ∩ (X − C). Ï� X − C ´X ¥�m8, �df�mÿÀ�½Â� B ´ Y ¥�m8. 5¿ B = Y −A, Ïd A ´ Y ¥�4

8. �

íØ 3.1.1 3ÿÀ�m X ¥, ek A ⊆ Y ⊆ X, � Y ´ X ¥�48, A ´ Y ¥�48,@o A �´ X ¥�48.

y² du A ´ Y ¥�48, d·K 3.1.1 , �3 X ¥�48 C, ¦� A = Y ∩ C

qÏ� Y,C Ñ´ X ¥�48, d48�5�, A = Y ∩ C �´ X ¥�48. �

e¡·�5½Â8Ü�SÜÚ4�.

½Â 3.1.1 X ´ÿÀ�m, Y ´ X �f8.

(1) Y �SÜInt(Y)

4=

⋃V⊆Y

V,

ùp V �H X ¥¤k÷v V ⊆ Y �m8. �ó�, Int(Y ) Ò´ X ¥¹u Y ��m8.k��Ö��B, �ò Y �SÜ{P� Y .

(2) Y (3 X ¥�) �4�.Y

4=

⋂C⊇Y

C,

ùp C �H X ¥¤k÷v C ⊇ Y �48. �ó�, Y Ò´ X ¥�¹ Y ��48. k�

�;�· , ·��^ ClX(Y ) L« Y �4�.

- 20 -


5 3.1.2 �·�!Ø Y �SÜÚ4��, �½�5¿§´��=��ÿÀ�m�f85½Â�. (�/`, Y �SÜÚ4��Vg�~�6u Y ¤?��µ�m. �

~ 3.1.1 �Ä X = R1 þ�IOÿÀ.

(1) � Y1 = (a, b). Y1 �SÜ Y1 = (a, b) = Y1, Y1 �4� Y 1 = [a, b].

(2) � Y2 = (a, b]. Y2 �SÜ Y2 = (a, b), Y2 �4� Y 2 = [a, b].

(3) � Y3 = [a, b]. Y3 �SÜ Y3 = (a, b), Y3 �4� Y 3 = [a, b] = Y3.

(4) � Y4 = {1}. Y4 �SÜ Y4 = ∅, Y4 �4� Y 4 = {1} = X − ((−∞, 1) ∪ (1,+∞)). �

~ 3.1.2 �Ä X = R2þ�IOÿÀ.

(1) � Y1 = (a, b)× (c, d), K

Y1 = (a, b)× (c, d) = Y1, Y 1 = [a, b]× [c, d].

(2) � Y2 = (a, b)× [c, d], K

Y2 = (a, b)× (c, d) = Y1, Y 2 = [a, b]× [c, d].

(3) � Y3 = {(x, y) | x, y ∈ Q}, KY3 = ∅, Y 3 = R2.

~ 3.1.3 E�Ä X = R1 þ�IOÿÀ.

(1) e Y = Z+ = {1, 2, 3, . . . }, K

Y = ∅, Y = Y = R1 − ((−∞, 1) ∪ (1, 2) ∪ · · · ∪ (n, n+ 1) ∪ · · · )

(2) e Y = { 1n | n ∈ Z+}, K

Y = ∅, Y = Y ∪ {0}.

(3) e Y = R+ = (0,+∞), KY = Y, Y = Y ∪ {0}.

(4) e Y = Q, K Y = ∅, Y = R1. �

5 3.1.3 ±eÛ��'uSÜÚ4��{ü5� (�ÖögC�y). � Y,A,B ´ X

�f8.

(1) Y ´ X ¥�m8 ⇐⇒ Y = Y ;

(2) Y ´ X ¥�48 ⇐⇒ Y = Y ;

(3) Y ⊆ Y ⊆ Y ;

- 21 -


(4) Y = Y , Int(Y ) = Y .

(5) e A ⊆ B, @o A ⊆ B, A ⊆ B. �

·K 3.1.2 (4��) � X ´ÿÀ�m, A, Y ©O´ X �f8, ÷v A ⊆ Y . � A ´

A 3 X ¥�4�, ClY (A) ´ A 3 Y ¥�4�, @o·�k

ClY (A) = A ∩ Y.

y² ·�ky ClY (A) ⊆ A∩Y . d·K 3.1.1, A∩Y ´ Y ¥�48, �w,k A ⊆ A∩Y .qÏ� ClY (A) ´ Y¥�¹ A ��48, ¤± ClY (A) ⊆ A ∩ Y .

2y A ∩ Y ⊆ ClY (A). d·K 3.1.1, ClY (A) = Y ∩ C, ùp C ´ X ¥,�48. Ï

A ⊆ ClY (A) = C ∩ Y ⊆ C.

d48�½Â� A ⊆ C. ù�,

A ∩ Y ⊆ C ∩ Y = ClY (A).

nþ, ·�y² ClY (A) = A ∩ Y . �

·K 3.1.3 (4��K) � X ´ÿÀ�m, Y ´ X �f8, B ´ X �ÿÀÄ, x ∈ X, @

o±eÃ^�*d�d:

(1) x ∈ Y ,

(2) éu X ¥?Û�¹ x �m8 U , Ñk U ∩ Y 6= ∅,

(3) éu B ¥?Û¹ x �Ä�� B, Ñk B ∩ Y 6= ∅.

5 3.1.4 8��Bå�, 3Ø· ��¹e, ·�r¹ x �m8¡� x ��. þã�

K�^� (2) ��±Qã�: x �?Û�� Y ��8��. �

y² (2) =⇒ (3) ù´²��, Ï� B ¥�Ä��´ X ¥�m8.

(3) =⇒ (2) ∀x ∈ X ±9 x �?¿�� U , dÿÀÄ�½Â, �3 B ∈ B ¦� x ∈ B ⊆ U .

db�^�, B ∩ Y 6= ∅, ¤± U ∩ Y 6= ∅.

±e·��Iy (1) Ú (2) ��d5.

(1) =⇒ (2) � x ∈ Y . b��3 x �,�� U , ÷v U ∩ Y = ∅, K Y ⊆ X − U . 5¿�X − U ´ X �48, �d4��½Â� Y ⊆ X − U . ùÒíÑ x ∈ X − U , = x 6∈ U , ù� U �À�gñ.

(2) =⇒ (1) b� x 6∈ Y , = x ∈ X − Y . 5¿� Y ´ X ¥�48, Ïd U = X − Y ´ x ��. �´U ∩ Y = ∅, ù�b�^�gñ. �

~ 3.1.4 � (X, d) ´Ýþ�m, B = Bd(x, ε) ´ ε-¥,

C = {y ∈ X | d(x, y) ≤ ε}.

- 22 -


(1) �y C ´48¿�

B ⊆ C.

Äk, é?Û y ∈ X − C, =÷v d(x, y) > ε, Ñ�3¥ Bd(y, δ) ÷v

Bd(y, δ) ∩B = ∅,

ùp δ = d(x, y)− ε. ù´Ï�, é?Û z ∈ Bd(y, δ),

d(x, z) ≥ d(x, y)− d(y, z) > d(x, y)− δ = ε.

Ïd X − C �±d¤kù«¥�¿��, ��m8, l C ´48. d4�½Â9 B ⊆ C, =�B ⊆ C.

(2) B �7�u C. 'X�Ä X �lÑÿÀ, §d~ 2.3.19 ¥�²�Ýþp�. � B =Bd(x, 1) = {x}. Ï� B �´48, ¤±§�4�Ò´g�. �´

C = {y ∈ X | d(x, y) ≤ 1} = X.

(3) éî¼�m X = Rn þ�IOÝþ, ·�k B = C. �

½Â 3.1.2 � X ´ÿÀ�m, Y ⊆ X, x ∈ X. e x �?Û�� Y ��8Ñ¹kÉu x

�:, K¡ x ´ Y �à: (Accumulation Point), �¡�4�:. �ó�, eéu�¹ x �?¿

m8 U ÑkU ∩ (Y − {x}) 6= ∅,

K x Ò´ Y �à:.

Y �¤kà:|¤�8Ü¡� Y ��8 (Derived set). Ï~P� Y ′.

5 3.1.5 |^4��K��, x ∈ Y ′ ��=�

x ∈ Y − {x}.

~ 3.1.5 �Ä X = R1 þ�IOÿÀ, � Y = (0, 1]. Ï� 0 ∈ [0, 1] = Y − {0}, � 0 ∈ Y ′

´ Y �à:. �

·K 3.1.4 � X ´ÿÀ�m, Y ⊆ X, K Y = Y ∪ Y ′.

y² (1) ky Y ′ ∪ Y ⊆ Y

Äk, e x ∈ Y ′, Kéu x �?¿�� U , U ∩ (Y − {x}) 6= ∅, Ïd U ∩ Y 6= ∅. d4��½Â, x ∈ Y . d x �?¿5, Ò�� Y ′ ⊆ Y . ,��¡, w,k Y ⊆ Y , Ï Y ′ ∪ Y ⊆ Y .

(2) 2y Y ⊆ Y ′ ∪ Y

� x ∈ Y , e x ∈ Y , Kw,k x ∈ Y ′ ∪ Y . 8Ø�� x 6∈ Y . d4��K, éu x �?¿�

� U , Ñk U ∩ Y 6= ∅. Ï� x 6∈ Y , � Y = Y − {x}, l U ∩ (Y − {x}) 6= ∅. ùÒíÑ x ∈ Y ′.Ïdd x �?¿5��, Y ⊆ Y ′ ∪ Y . �

5 3.1.6 k�·��¬^�f�m Y �“>.”ù�Vg. §½Â�

Bd(Y ) = Y ∩X − Y .

- 23 -


íØ 3.1.2 � X ´ÿÀ�m, Y ⊆ X, @o±e^�*d�d:

(1) Y ´48;

(2) Y = Y ;

(3) Y ′ ⊆ Y .

y² (1)=⇒(2) w,.

(2)=⇒(3) Ï� Y = Y = Y ∪ Y ′, ¤± Y ′ ⊆ Y .

(3)=⇒(1) ��¡, ·�k Y ⊆ Y . ,��¡, db�^�, Y = Y ′ ∪ Y ⊆ Y , Ïd Y = Y ,u´ Y ´4�. �

·K 3.1.5 � X ´ÿÀ�m, B ⊆ A ⊆ X. eP IntA(B) L« B 3 A ¥�SÜ. @oIntA(B) = A−A−B. AO/, Int(B) = X −X −B.

y² 5¿

A−A−B = A ∩ (X −A−B),

Ï §´ A ¥�m8. d,

A−A−B ⊆ A− (A−B) = B.

Ï� IntA(B) ´ A ¥�¹8Ü B ��m8, ¤± A−A−B ⊆ IntA(B).

e¡, ·��y² IntA(B) ⊆ A−A−B, =Iy IntA(B) ∩A−B = ∅. 5¿� IntA(B) ∩(A−B) = ∅, ¤±·��I�y² IntA(B) ∩ (A−B)′ = ∅. �é{`, éz� x ∈ IntA(B), ·

�I�é� x (3 X ¥) �� U ¦� U ∩ (A−B) = ∅.

Ï� IntA(B) ´ A ¥�m8, ¤±�3 X ¥�m8 U , ¦� IntA(B) = U ∩A. Ï�

U ∩ (A−B) = (U ∩A) ∩ (X −B) = IntA(B) ∩ (X −B) ⊆ B ∩ (X −B) = ∅,

¤± U ∩ (A−B) = ∅. éz� x ∈ IntA(B), U ´ x ��, §÷v·�I��^�. �

3.2 Hausdorff 5�

½Â 3.2.1 � X ´ÿÀ�m, eéu X ¥?¿ü�ØÓ�: x, y, o�3 x �� U Ú

Y �� V , ÷v U ∩ V = ∅, K¡ X ´ Hausdorff �m, ½` X (3TÿÀe) ´ Hausdorff �.

~ 3.2.1 �Äî¼Ýþ�m X = Rn þ�IOÿÀ. ùw,´�� Hausdorff �m. �

~ 3.2.2 �Ä?�8Ü X þ�lÑÿÀ. d� X ´ Hausdorff �. ù´Ï�, é?¿ü�ØÓ�: x, y ∈ X, x ∈ {x}, y ∈ {y}, � {x}, {y} w,´Ø��m8. �

~ 3.2.3 �Än�î¼�m¥�¥¡

S2 = {(x, y, z) ∈ R3|x2 + y2 + z2 = 1}.

§�� R3 þ�f�mÿÀ�´ Hausdorff�. �

- 24 -


~ 3.2.4 X = {1, 2, 3}, T = {∅, X, {2, 3}, {1, 2}, {2}}, ·�� x = 1, y = 3. �¹ x

�m8=k X, {1, 2}; �¹ y �m8k:X, {2, 3}. dd��, x Ú y ��o´��. Ïdù�ÿÀ�mØ´ Hausdorff �. �

~ 3.2.5 �Ä�� X = R1 þ�{k�ÿÀ

T4= {U | X − U ´k�8½ö U ´�8}.

§Ø´ Hausdorff �.

¯¢þ, éu X ¥?¿ØÓ�ü�� x, y, 1e�3 x �� U Ú y �� V , ¦�U ∩ V = ∅, Kdúª:

X = (X − U) ∪ (X − V )

X − U Ú X − V Ñ´k�:8, l X �´k�:8, ùÒ��gñ! �

~ 3.2.6 � X = Cn, T ´ X þ� Zariski ÿÀ (�~ 2.2.6). (X,T ) Ø´ Hausdorff �m. Öö�±aqþ~��{�yù�(Ø. �

·K 3.2.1 � X ´äkSÿÀ��S8, K X ´ Hausdorff �m.

y² � x, y ∈ X, x 6= y, Ø�� x, y Ø´�� (�) �. d��é�� z, w , ¦�z < x, y < w. ·�©�/?Ø:

(1) b� x � y ��, = (x, y) = ∅.

- U = (z, y) ,V = (x,w) w,, U Ú V Ñ´ X ¥�m8, � x ∈ U , y ∈ V . ·�k

U ∩ V = (z, y) ∩ (x,w) = (x, y) = ∅.

(2) b� x � y Ø��, =�3 h ∈ (x, y).

- U = (z, h) ,V = (h,w). d� x ∈ U , y ∈ V , �

U ∩ V = (z, h) ∩ (h,w) = ∅.

aq/, ·��±?Ø x, y ´�� (�) ��/. d?Ø2Kã. �

·K 3.2.2 � X Ú Y ´ Hausdoaff �m, K X × Y �´ Hausdorff �m.

y² � a× b, c× d ∈ X × Y , ·�E,©�/?Ø.

(1) b� a 6= c � b 6= d.

Ï� X Ú Y ©OÑ´ Hausdorff, Ïd·�Ué� X ¥Ø��m8 U1, U2, ±9 Y ¥

Ø��m8 V1, V2, ¦�

a ∈ U1, b ∈ V1, c ∈ U2, d ∈ V2.

Ïd

(U1 × V1) ∩ (U2 × V2) = (U1 ∩ U2)× (V1 ∩ V2) = ∅.

- 25 -


(2) b� a = c ½ b = d.

·�Ø�� a = c, ,��/�aq?Ø. d X Ú Y � Hausdorff 5�, ·�E,�±é�a �� U ±9 b �� V1, d �� V2, ¦�:

a× b ∈ U × V1, c× d ∈ U × V2

¿� V1 ∩ V2 = ∅, Ïd

(U × V1) ∩ (U × V2) = U × (V1 ∩ V2) = ∅.

ùÒy²(Ø. �

·K 3.2.3 X ´ Hausdorff, Y ⊆ X. K Y ��f�mÿÀ´ Hausdorff.

y² éu?¿ x, y ∈ Y, x 6= y, Ï� X ´ Hausdorff, ¤±�3 X ¥�Ø��m8

U, V ¦� x ∈ U, y ∈ V , u´

x ∈ Y ∩ U, y ∈ Y ∩ V.

d,(Y ∩ U)× (Y ∩ V ) = Y × (U ∩ V ) = ∅.

Ïd, d x, y À��?¿5, Y ��f�mÿÀ´ Hausdorff �. �

~ 3.2.7 ·�òü^��þ,Ø�:, �Ó ��:©OÊÜ¤�:. ù��8

Üäk�k��þ�g,�ÿÀ. §Ú��O==3u: cökü�ØÓ��:, �ö=k��:. �ó�, T8Ü�±w¤´ò�^��:©�¤ü�:.

ù�ÿÀ�mØ´ Hausdorff. Ï�§�ü��:Ã{^Ø��«©m. 3�©AÛ¥, ·�?Ø6/�, o´�b� Hausdorff ^�¤á. ù��Ï��, Ò´�üØù«Û

%�AÛ~f. �

½n 3.2.1 (Hausdorff �m�K) � X ´ÿÀ�m, K±eü�^�*d�d:

(1) X ´ Hausdorff �m.

(2) X ×X �é�� 4 = {(x, x) | x ∈ X} ´ÈÿÀe�48.

3y²ù��Kc, ·�I��O�ó�.

Ún 3.2.1 X ´ÿÀ�m, � U, V ´ X ¥�m8, � 4 = {(x, x) | x ∈ X}

K±e^��d:

(1) U ∩ V = ∅

(2) (U × V ) ∩4 = ∅

(3) U × V ⊆ (X ×X)−4

- 26 -


y² (2)⇐⇒(3) w,.

(1)=⇒(2) ^�y{. � (U × V ) ∩ 4 6= ∅, @o�3 (x, x) ∈ 4 � (x, x) ∈ U × V . u´x ∈ U ∩ V , ù�^�gñ.

(2)=⇒(1) ^�y{. � U ∩ V 6= ∅, @o�3 x ∈ U ∩ V , u´ (x, x) ∈ (U × V ) ∩4, ¤±

(U × V ) ∩4 6= ∅, ù�^�gñ. �

y3·�5y²½n 3.2.1.

y² (1)=⇒(2) �y 4 ´48, ��y² X ×X −4 ´m8.

éu?¿� (x, y) ∈ X ×X −4 , ·��I�é� (x, y) �� U × V ¦�

U × V ⊆ X ×X −4,

=

(U × V ) ∩4 = ∅.

dÚn 3.2.1, ù�dué x �� U 9 y �� V , ¦� U ∩ V = ∅. ^� (1) w,�yù�

��35.

(2)=⇒(1) � x, y ∈ X,x 6= y. Ï� 4 ´ X ×X ¥�48, ¤± X ×X −4 ´m8. ·�Ué� X ¥�ü�m8 U, V , ¦�:

(x, y) ∈ U × V ⊆ X ×X −4.

�âÚn 3.2.1, U ∩ V = ∅ , x ∈ U , y ∈ V . ùÒy² X ´ Hausdorff �. �

·K 3.2.4 � X ´ Hausdorff �m, Y ´ X ¥�k�:8, K Y ´48.

y² Ï�k��48�¿E,´48, ¤±·��Iy² Y ´ü:8��/.

� Y = {x}. �y Y ´48, �Iy X − {x} ´m8. ?� y ∈ X − {x}, �Ié� y ��

�¹u X − {x} ¥=�.

Ï� X ´ Hausdorff �m, ¤±�3m8 U, V ÷v x ∈ U , y ∈ V , U ∩ V = ∅. Ï V ⊆ X − {x}. �

·K 3.2.5 � X ´ Hausdorff �m, Y ⊆ X, x ∈ X, K±e^��d:(1) x ∈ Y ′,(2) x �?Û��¹k Y ¥Ã¡õ�:.

y² (2) =⇒ (1) du x ¥�?Û��¹k Y ¥Ã¡õ�:, Ïd¹kÉu x �:. Ïd

x ´ Y �à:.

(1) =⇒ (2) ®� x ∈ Y ′. ·�æ^�y{. b��3 x �,�� U , ÷v

U ∩ (Y − {x}) = {p1, p2, · · · , pm}.

-

V = U − {p1, p2, · · · , pm} = (X − {p1, p2, · · · , pm}) ∩ U.

Ï� X ´ Hausdorff �, �â·K3.2.4, X − {p1, p2, · · · , pm} ´m8, Ïd V �´ X ¥�m8,¿�w,k x ∈ V .

- 27 -


·��Iy² V ∩ (Y − {x}) = ∅, ù�Ò� x ∈ Y ′ gñ. ¯¢þ,

V ∩ (Y − {x}) = ((X − {p1, p2, · · · , pm}) ∩ U) ∩ (Y − {x})= (X − {p1, p2, · · · , pm}) ∩ ({p1, p2, · · · , pm})= ∅.

dd(Ø�y. �

3.3 ëÏ5

½Â 3.3.1 � X ´ÿÀ�m. e�3ü�Ø��m8 U, V ÷v U ∪ V = X, K¡X ´ØëÏ� (½¡�©��), þã U, V ¡� X ��©�. eØ�3þãm8 U, V K¡ X

´ëÏ�.

5 3.3.1 Uì½Â, X ´ëÏ��=�Ø�3 X þ��m8 U, V , ÷v U ∩ V = ∅9 U ∪ V = X. �

·K 3.3.1 � X ´ÿÀ�m, K±eü�^�*d�d:(1) X ´ëÏ�;(2) X ¥Ø ∅ Ú X Ø�3Ù¦f8ÜQ´m8q´48.

y² (1)=⇒(2) �y{. b��3ù��f8Ü U , Q´m8�´48, � U 6= ∅, X. -

V = X − U , @o V 6= ∅, X �÷v {U ∩ V = ∅U ∪ V = X

Ï� U ´48, ¤± V ´m8. ù� U Ú V ´ X ��©�, = X ØëÏ, �^�gñ!

(2)=⇒(1) �y{. b� X ØëÏ, � U Ú V ´ X ��©�, KÏ� V ´ X ¥�m8�� U = X − V ´48, ¿� U 6= ∅, X, gñ! �

~ 3.3.1 �Ä X = R1 þ�IOÿÀ. d� X ´ëÏ�. ù�¯¢¿�w,, ·��¡ò

�y²§. �

~ 3.3.2 � (X,T ) ´lÑÿÀ, ��¹kü��. éu X ¥?¿�� x, Ï�ü

:8 {x} Q´m8�´48, �k

X = {x} ∪ (X − {x})X ∩ (X − {x}) = ∅

¤± {x} Ú X − {x} ´ X ��©�, = X ØëÏ. �

~ 3.3.3 (X,T ) ´²�ÿÀ, T = {∅, X}, w, X ´ëÏ�. �

~ 3.3.4 X = R1, T ´{k�ÿÀ, =

T = {U | U = ∅ ½ X − U ´k�:8}

K (X,T ) ´ëÏ�.

- 28 -


·�5�yd(Ø. b�,m8 U 6= ∅, X, ¿� U �´48, @o X − U �´m8, �é{5`, X − (X − U) = U ´k�8, Ï� U ´m8, ¤±X − U �´k�8, u´

X = U ∪ (X − U)

�´k�8, gñ! ÏdØ�3Ø�8Ú�8Q´m8�´48�f8Ü, ¤± (X,T ) ´ëÏ�.

,, ��[5¿, (X,T ) ¿Ø´ Hausdorff �. �

~ 3.3.5 �Ä X = R1 þ�IOÿÀ. � Y = (0, 1) ∪ (2, 3), @o (0, 1) Ú (2, 3) ´f�m

Y ��©�.

ù�~fL², ��mëÏ, Ùf�mØ�½ëÏ. �ó�, ëÏ5ù�ÿÀ5�¿ØU¢D

�f�m. ØL·K 3.2.3 L², Hausdorff 5�´U¢D�f�m�. �

·K 3.3.2 (f�m�©�5�K) � X ´ÿÀ�m, Y ⊆ X, @o±e^�*d�d:(1)Y ØëÏ;(2)�3��f8 A,B ⊆ Y ÷v{

A ∪B = Y,

A ∩B = ∅,

{A ∩B′ = ∅,A′ ∩B = ∅,

ùp A′, B′ ©O� A,B 3 X ¥��8.(3) �3 Y ��f8 A,B ÷v

A ∪B = Y,

A ∩B = ∅,A ∩B = ∅,

ùp A Ú B ©O� A,B 3 X ¥�4�.

±þ^��¤á�, A,B Ò´ Y ��©�. �L5, XJ X ØëÏ, @o§�?�©

�Ñ÷vþã^�.

y² (1)=⇒(2) ®� Y ´ØëÏ�, ·�� Y ¥��©� A,B. §�� Y ¥��m8, w,÷v {

A ∪B = Y

A ∩B = ∅

Ï� A �´ Y ¥�48, ¤±

A = ClY (A) = A ∩ Y.

ù�, ·�k

A = A ∩ Y = A ∩ (A ∪B) = (A ∩A) ∪ (A ∩B) = A ∪ (A ∩B).

qÏ�

(A ∩B) ∩A = (A ∩A) ∩B = A ∩B = ∅,

¤± A ∩B = ∅. d A = A′ ∪A ��

A′ ∩B ⊆ A ∩B = ∅,

- 29 -


l A′ ∩B = ∅. Ón A ∩B′ = ∅. Ïd A,B ÷v¤I^�.

(2)=⇒(1) � A,B ´÷v^� (2) ��f8, u´k

A ∩B = ∅, A ∩B′ = ∅

(Ü B = B′ ∪B �� A ∩B = ∅. Ón, A ∩B = ∅. ù�Òk

(A ∩ Y ) ∩B = (A ∩B) ∩ Y = ∅.

,��¡,Y = A ∪B = (A ∩ Y ) ∪B ⊆ (A ∩ Y ) ∪B ⊆ Y

%¹X Y = (A ∩ Y ) ∪B. ùÒk B = Y − (A ∩ Y ). Ïd B ´ Y ¥�m8. Ón, A �´ Y ¥

�m8. Ïd, A Ú B ´ Y ��©�, Y ØëÏ.

(2)=⇒(3) w,.

(3)=⇒(2) d^��, A ∩B ⊆ A ∩B = ∅, = A ∩B = ∅.

2�â (A′ ∪A) ∩B = A ∩B = ∅, íÑ A′ ∩B = ∅. Ón�y A ∩B′ = ∅. �

·�Þ�k'f�mëÏ5�~f.

~ 3.3.6 � (R1,T ) ´IOÿÀ, Y1, Y2 ´ X �f8.

(1) � Y1 = [−1, 0) ∪ (0, 1]. A = [−1, 0) ±9 B = (0, 1] ´ Y1 ��©�.

ù´Ï�·��±rþãü�8Ü�¤:

A = Y1 ∩ (−2, 0), B = (0, 1] = Y1 ∩ (0, 2).

du (−2, 0), (0, 2) Ñ´ X ¥�m8, Ïd A,B ´ Y1 ¥�Ø��m8, ��8´ Y1. Ï

d Y1 ´ØëÏ�.

·��±|^f�m�©�5�K5�yù�¯. ù´Ï� A ∪B = Y1, ¿�

A ∩B = [−1, 0] ∩ (0, 1] = ∅,

A ∩B = [−1, 0) ∩ [0, 1] = ∅.

d�Ká� A Ú B ´ Y1 ��©�.

(2) � Y2 = [−1, 1] = [−1, 0] ∪ (0, 1]. P A = [−1, 0], B = (0, 1]. ·�5`² A Ú B Ø´ Y2

��©�. ¯¢þ, B = [0, 1],Ï A ∩ B = {0} 6= ∅. df�m©��K�� A,B Ø´ Y2 �©�. �¡·�òy² Y2 ´ëÏf8. �

~ 3.3.7 (R1,T ) ´IOÿÀ, Y = Q ´knê8. � Y ¥��m8

A = Y ∩ (−∞,√

2), B = Y ∩ (√

2,+∞),

K A,B ´ Y ��©�.

?�Ú, XJ A ´ Y ¥��f8, � A ¥��¹kü��, ·��yf8 A ØëÏ.� p, q ∈ A, �Ø�� p < q. aqþ¡��{, ��Ãnê α ∈ (p, q), ¿� A ¥m8

U = A ∩ (−∞, α), V = A ∩ (α,+∞).

U, V w,÷v U ∩ V = ∅ 9 U ∪ V = A. Ïd U, V ´ A ��©�. �

- 30 -


~ 3.3.8 (R2,T ) ´IOÿÀ,

Y = {(x, y) | y = 0} ∪ {(x, y) | y =1x

� x > 0}

´ R2 �f8, eP

A = {(x, y) | y = 0}, B = {(x, y) | y =1x

� x > 0},

@o A,B ´ X ¥Ø��48. d©�5�K, §�´ Y ��©�, Ï Y Ø´ëÏ�. �

Ún 3.3.1 � X ´ØëÏ�, A,B ´ X ��©�, Y ⊆ X ´ëÏf8, K�o Y ⊆ A,�o Y ⊆ B.

y² Ï� A,B Ñ´ X ¥�m8, ¤± A ∩ Y,B ∩ Y ´ Y ¥�m8, ÷v{(A ∩ Y ) ∪ (B ∩ Y ) = Y,

(A ∩ Y ) ∩ (B ∩ Y ) = ∅.

e A∩Y � B∩Y Ñ´��, K§�´ Y �©�, ù� Y �ëÏ5gñ! Ïd, �o A∩Y = ∅,�o B ∩ Y = ∅. �ó�, �o A ∩ Y = Y , �o B ∩ Y = Y . ù��u`, �o Y ⊆ A, �oY ⊆ B. �

·K 3.3.3 (ëÏf8�¿) � {Yα}α∈I ´ X ¥��xëÏf8. e⋂

α∈I

Yα 6= ∅, K⋃

α∈I

Yα

�ëÏ.

y² du⋂

α∈I

Yα 6= ∅, ��: p ∈⋂

α∈I

Yα. � Z =⋃

α∈I

Yα. b� Z = A∪B ´��©�,

dÚn 3.3.1 �, éz� Yα, �o Yα ⊆ A, �o Yα ⊆ B. ·�Ø�� p ∈ A, @oéu?¿

� α ∈ I, Ñk Yα ⊆ A, ddíÑ A = Z, B = ∅. ù�b�^�gñ! �⋃

α∈I

Yα ´ëÏ8. �

·K 3.3.4 (ëÏf8�4�) X ´ÿÀ�m, Y ⊆ X ´ëÏf8, Z ´ X ��f8,÷v

Y ⊆ Z ⊆ Y .

@o Z �´ëÏf8. AO/, ëÏf8�4�E´ëÏ�.

y² b� A,B ´ Z ��©�. Ï� Y ⊆ Z, �âÚn 3.3.1, �o Y ⊆ A, �oY ⊆ B. Ø�� Y ⊆ A, l Y ⊆ A.

,��¡, df�m©�5�K, A ∩B = ∅. Ïd

B = Z ∩B ⊆ Y ∩B ⊆ A ∩B = ∅

%¹X B = ∅, ù�b�^�gñ! �

½n 3.3.1 (ëÏ�m�¦È) � X,Y ´ëÏ�m, @o X × Y �´ëÏ�.

y² 8�E X × Y ��xf8

Tx = (X × b) ∪ (x× Y ),

- 31 -


ùp b ∈ Y ´,��½:, x �H X ¥�:.

u´ ⋃

x∈X

Tx = X × Y⋂x∈X

Tx = X × b.

XJUy² Tx ëÏ, @od·K 3.3.3 á� X × Y �ëÏ5.

e¡y² Tx ëÏ. 5¿�,

(x× Y ) ∩ (X × b) = {(x, b)} 6= ∅,

Ïd·��Iy² x× Y Ú X × b Ñ´ëÏ�, ? d·K 3.3.3 �íÑ Tx ´ëÏ�.

Ø��Ä X × b. b� X × b k©� A,B. ·��

A = A0 × b B = B0 × b,

ùp� A0, B0 ´ X ¥Ø��m8. N´�y, A0, B0 ´ X ��©�, ù�Ò� X �ëÏ5gñ! � X × b ´ëÏ�. aq�y, x× Y ´ëÏ�. �

íØ 3.3.1 � X1, X2, · · · , Xn ´ëÏ�m, K X1 ×X2 × · · · ×Xn �´ëÏ�.

5 3.3.2 ¯¢þ, ·��±y²��(Ø: � {Xα}α∈I ´ëÏ�mx, K∏α∈I

Xα ��

ÈÿÀ�´ëÏ�, �§��ÿÀ�7ëÏ. �

·�£�3�!�m©¤Þ�~f,=�kIOÿÀ�¢�� X = R1 ±9Ù¥�m«m½ö´m��. ·��î�y²§�3IOÿÀe´ëÏ�. ù�(Ø�y²¿Ø´w,�. e¡·�ò?Ø��/.

½Â 3.3.2 � X ´�S8. e X ÷v±e^�:(1)(þ(.5�) ?Ûk.f8Ñkþ(.;(2)(0�5) ∀x, y ∈ X,÷v x < y, @o ∃z ∈ X, ¦� x < z < y,·�Ò¡ X ´�5ëYÚ.

½n 3.3.2 (�5ëYÚ�ëÏ5) � X ´�5ëYÚ, K X ´ëÏ8, ¿� X �m«m

±9m��ëÏ.

AO�, IOÿÀe�¢��±9Ùþ�m«m½m��Ñ´ëÏ�.

y² � X ´�5ëYÚ, Y ´ X ¥�m«m½m��½ X ��. ·��8I´y²ù�� Y ´ëÏ�.

æ^�y{. � A,B ´ Y ��©�. � a ∈ A, b ∈ B �Ø�� a < b. d Y �À�, ·�k [a, b] ⊆ Y . -

A0 = A ∩ [a, b], B0 = B ∩ [a, b].

w, A0, B0 ´ [a, b] ��©�. ·�P A0 �þ(.� c = supA0. Ï� c ∈ A0 ∪ B0, ��

oc ∈ B0 �o c ∈ A0. ·�y²ùü«�¹Ñ´Ø�U�, l �Ñgñ.

- 32 -


(1) b� c ∈ B0. Ï� B0 ´ [a, b] ¥�m8� c 6= a, ¤±�3 d ∈ B0, ¦� (d, c] ⊆ B0. 5¿

�, d��o c = b, �o a < c < b. e c = b, @o d w,�´ A0 �þ.. � d < c, ù� c

´þ(.gñ! Ïd c < b. Ï� c ´ A0 �þ., ¤± A0 ∩ (c, b] = ∅, u´

(d, b] = (d, c] ∪ (c, b] ⊆ B0.

ùL² d ´ A0 �þ., EÚ c ´þ(.gñ!

(2) b� c ∈ A0. d� c 6= b, u´�o a < c < b, �o c = a. Ï� A0 ´m8, ¤±�3e ∈ A0, ¦� [c, e) ⊆ A0. d0�5^�, �3 z ∈ (c, e), l z ∈ A0. ù� c ´8Ü A0 �þ.gñ!

nþ��, Y ´ëÏ�. �

� X ´ÿÀ�m, x, y ∈ X. ·�½Â X ¥��'X ∼:

x ∼ y ⇐⇒ x, y á3,�ëÏf8S.

Ún 3.3.2 þã�'X ∼ ´�d'X.

y² ·��yþã� ∼ ÷v�d'X�n�^�:

(1) g�5: Ï� x ∈ {x} � {x} ´ëÏf8, ¤± x ∼ x;

(2) é¡5: e x ∼ y, @o x Ú y á3��ëÏf8¥, Ï �k y ∼ x;

(3) D45: � x ∼ y, y ∼ z. Ø�� x, y Ñ3ëÏf8 A ¥, y, z Ñ3ëÏf8 B ¥. -

C = A ∪B, Ï� y ∈ A ∩B, �d·K 3.3.3, C �´ëÏ�. d x, z ∈ C íÑ x ∼ z. �

½Â 3.3.3 d ∼ ½Â��da¡� X �ëÏ©|.

5 3.3.3 X ´ëÏ��=� X =k��ëÏ©|. �

·K 3.3.5 � X ´ÿÀ�m, K

(1) X �ëÏ©|´ëÏf8.(2) �ëÏ©|*dØ��, Ù¿8� X.(3) ?�ëÏf8=¹3��ëÏ©|p.

y² (1) ?� x0 ∈ A, ùp A ´ëÏ©|, éu?¿� x ∈ A, Ï� x ∼ x0, ¤±�3ëÏf8 Ax, ¦� x, x0 ∈ Ax. Ï� Ax ⊆ A, ¤±

⋃x∈A

Ax = A. ,��¡, w,k x0 ∈⋂

x∈A

Ax. u

´ ⋃

x∈A

Ax = A,⋂x∈A

Ax 6= ∅.

d·K 3.3.3 � A ´ëÏ�.

(2) 5gu�da�½Â.

- 33 -


(3) � Y ´ëÏf8. æ^�y{. b� Y �ü�ØÓ�ëÏ©| A,B ��. �

x ∈ A ∩ Y, y ∈ B ∩ Y.

Ï� x, y ∈ Y , ¤± x ∼ y, u´ x, y á3Ó��ëÏ©|¥, l A = B, gñ! Ïd Y ��

�ëÏ©|��, l Y ¹uT©|¥. �

~ 3.3.9 � X = [−1, 0) ∪ (0, 1] ⊆ R1 ´IOÿÀe�f�m, u´ X kü�ëÏ©|,©O´ [−1, 0) Ú (0, 1]. �

~ 3.3.10 � X = Q ⊆ R1 ´IOÿÀe�f�m. d� X �ëÏ©|´ü:8 {x}, ùp x �H Q ¥��. �

3.4 ;�5

½Â 3.4.1 � X ´ÿÀ�m, A ´ X �f8x,(1)e A �¤k��¿�u X, K¡ A CX X, ½¡ A ´ X ��CX.(2)e A ´ X ��CX, � A ¥��m8, K¡ A ´ X �mCX.

~ 3.4.1 � X = {1, 2, 3}, T = {∅, X, {1, 2}, {2, 3}, {2}} ´ X þ�ÿÀ.(1) A1 = {{1}, {2}, {3}} ´ (X,T ) �CX. Ï� {1}, {3} ÑØ´m8, ¤± A1 ¿Ø´ X �mCX.(2) A2 = {{1, 2}, {2, 3}} ´ (X,T ) �CX, ¿� A2 ¥��Ñ´ X �m8, Ïd A2 ´ X �mCX. �

~ 3.4.2 �Ä X = Rn þ�IOÿÀ, B = { ¤km�� }, @o B ´ X �mCX. �

½Â 3.4.2 e X ¥�?ÛmCX A ¥o�¹��k��fx A ′ ⊆ A , ¦� A ′ �´ X

�mCX, K¡ X ´;�½;�� (Compact).

~ 3.4.3 �Ä X = R1 þ�IOÿÀ, A = {(n, n + 2) | n ∈ Z} ´ R1 �mCX. A ¥k��m«m�¿w,Ø�U�u X, � X Ø´;�. ��/, î¼�mÑØ´;8. �

~ 3.4.4 3 X = R1 �IOÿÀe�Ä X ¥�f�m Y1 = [0, 1], Y2 = (0, 1), Y3 = (0, 1).

(1) Y1 ´;8, ù5 uêÆ©Û¥�y². ·�ò3�©y²��(Ø.

(2) Y2 Ø´;8, �Ä8x A = {( 1n , 1) | n ∈ Z}, §´ Y2 �mCX, � A ¥�?Ûk�f8

xÑØ�UCX Y2.

(3) Y3 �Ø´;8, �aq�Ä Y3 �mCX B = {( 1n , 1] | n ∈ Z}. �

~ 3.4.5 � X = {1, 2, · · · , n},K X 3?ÛÿÀe��;8. �

½Â 3.4.3 � Y ´ X �f8, A ´ X �f8x, XJ A ¥��¿�¹ Y , K¡ A C

X Y .

- 34 -


·K 3.4.1 (f�m;5�K) � Y ⊆ X, K±e·K�d:

(1) Y ��f�mÿÀ´;�;

(2) d X ¥m8¤|¤� Y �?¿CXÑ�¹��k�fxCX Y .

y² (1)=⇒ (2) ®� Y ´;8. ?� X ¥�m8x A = {Uα}α∈I . ·��8I´�é

� X ¥k��m8 U1, U2, · · · , Un ∈ A ¦� Y ⊆n⋃

i=1Ai. �Ä Y �mCX

AY = {Uα ∩ Y | α ∈ I}.

�â Y �;5, �±é�

U1 ∩ Y, U2 ∩ Y, · · · , Un ∩ Y ∈ AY ,

÷vn⋃

i=1(Ui ∩ Y ) = Y . dd�

Y ∩ (n⋃

i=1

Ui) =n⋃

i=1

(Ui ∩ Y ) = Y

= Y ⊆n⋃

i=1Ui.

(2)=⇒ (1) ·��y Y ´;�, =y: XJ A ′ = {U ′α}α∈I ´ Y �?¿mCX,@o�3k

��mCX U ′1, U′2, · · · , U ′n ∈ A ′ ¦�

n⋃i=1

U ′i = Y .

Ï� U ′α ´ Y ¥�m8, ¤±�3 X ¥�m8 Uα ¦� U ′α = Uα ∩ Y . qÏ�⋃

α∈I

Uα′ = Y

Ïd

Y =⋃α∈I

(Uα ∩ Y ) = Y ∩ (⋃α∈I

Uα)

%¹ Y ⊆⋃

α∈I

Uα. - {Uα}α∈I = A , @o A ´ X ¥�m8x¿�CX Y .

d^� (2) , �3 X ¥�k�fx U1, U2, · · · , Un ∈ A ¦� Y ⊆n⋃

i=1Ui, = Y =

n⋃i=1

U ′i . �

~ 3.4.6 (1) � X = [0, 1], Y = (12 ,

34) ⊆ X. �, X ´;8, � Y �� X �f�m´�

;��.(2) � X = R1, Y1 = (0, 1), Y2 = [0, 1] Ñ´ X �f8. ÿÀ�m X Ø´;8, Ùf8 Y1 �Ø´;8, �f8 Y2 ´;�.

ùü~L², XJØ\·��^�, @oÿÀ�m�;5�Ùf�m�;5¿Ã7,éX.�

·K 3.4.2 � X ´;�m, Y ´ X �4f8, @o Y �´;8.

y² � A ´ X ¥�?¿CX Y �m8x.d·K 3.4.1, �Iy² A ¥��3k�fxCX Y . 5¿ Y ´48, � X − Y ´ X ¥�m8. ½Â

A ′ 4= A ∪ {X − Y }

ù´ X �mCX, = ⋃Uα∈A

Uα ∪ (X − Y ) = X.

- 35 -


d X �;5, �é� X ¥k�CX U1, U2, · · · , Un ∈ A ,¦�

(n⋃

i=1

Ui) ∪ (X − Y ) = X,

l Y ⊆n⋃

i=1Ui. d A �?¿5, Y �´;8. �

·K 3.4.3 � X ´ Hausdorff �, Y ´ X ¥�;f8, K Y ´48.

y² ·��y X − Y ´m8, =é X − Y ¥?�: x, �é�m8 U ¦� x ∈ U ⊆X − Y .

?� Y ¥��: y. Ï� X ´ Hausdorff �, ��3 X ¥�m8 Uy, Vy ÷v

x ∈ Uy, y ∈ Vy, Uy ∩ Vy = ∅.

4 y �H Y ¥�z�:, @o·�� X �m8x A = {Vy}y∈Y , §CX Y .

d Y �;59·K 3.4.1, �é� Vy1 , Vy2 , · · · , Vyn∈ A , ¦� Y ⊆

n⋃i=1

Vyi. -

U = Uy1 ∩ Uy2 ∩ · · · ∩ Uyn.

§´ X ¥�m8¿�w,�¹ x. Ï�

U ∩ Vyi= (

n⋂i=1

Uyi) ∩ Vyi

⊆ Uy1 ∩ Vy1 = ∅

¤± U ∩ Vyi= ∅, l U ∩ Y = ∅. ddíÑ x ∈ U ⊆ X − Y . �

·��±lþã·K�y²¥��Xek^�©l5(Ø.

íØ 3.4.1 � X ´ Hausdorff�, Y ´ X �;f8, x ∈ X − Y , K�3 X ¥Ø��m

8 U, V ¦� x ∈ U, Y ⊆ V .

½n 3.4.1 k�õ�;�m�È´;�. �ó�, � X1, X2, · · · , Xn ´;�m, K X1 ×X2 × · · · ×Xn �´;�.

3y²T½nc, ·�k�ÑXeÚn.

Ún 3.4.1 (+/Ún) � X,Y ´ÿÀ�m, Y ´;�, x0 ∈ X, W ´ X × Y ¥�¹

x0 × Y �m8, K�3 X ¥�m8 U , ¦� x0 ∈ U � U × Y ⊆W .

y² ?� (x0, y) ∈ x0×Y , �3 X×Y ¥�Ä�� Uy×Vy, ¦� (x0, y) ∈ Uy×Vy ⊆W .ùÒ�Ñ Y �mCX {Vy}y∈Y . �â Y �;5, �3k��m8

Vy1 , Vy2 , · · · , Vyn

¦� Y = Vy1 ∪ Vy2 ∪ · · · ∪ Vyn. w�

x0 × Y ⊆ (Uy1 × Vy1) ∪ (Uy2 × Vy2) ∪ · · · ∪ (Uyn× Vyn

) ⊆W.

- U = Uy1 ∩ Uy2 ∩ · · · ∩ Uyn, w, x0 ∈ U . ·��y²þã� U Ò´·��é� X ¥m

8, =÷v U × Y ⊆W .

- 36 -


?� U ×Y ¥�� (x, y), Ï� (x0, y) ∈ x0×Y , ¤±�3 Vyi¦� (x0, y) ∈ Uyi

×Vyi, l

y ∈ Vyi. u´

(x, y) ∈ U × Vyi⊆ Uyi

× Vyi⊆W.

ùÒíÑ U × Y ⊆W . �

y3·�5y²½n 3.4.1.

y² ·��Äü�;�m�¦È X × Y ��/. ��/�d8B{��.

� A ´ X × Y �?�mCX. ?� X �� x, Ï� x× Y ´;��, ¤±�3 A ¥k�

�� A1, A2, · · · , An ∈ A ¦� x× Y ⊆n⋃

i=1Ai. d+/Ún, �3 X ¥�m8 Wx, ¦�

x× Y ⊆Wx × Y ⊆ A1 ∪A2 ∪ · · · ∪An,

= Wx×Y �� A ¥k��CX. éu X ¥?¿�: x, Ñ�3þã X ¥�m8 Wx. Ïd

{Wx}x∈X ´ X ��mCX. Ï� X ´;�m, ¤±�3 Wx1 ,Wx2 , · · · ,Wxn∈ {Wx}x∈X ¦�

X = Wx1 ∪Wx2 ∪ · · · ∪Wxn

�Ò´`

X × Y = (Wx1 × Y ) ∪ (Wx2 × Y ) ∪ · · · ∪ (Wxn× Y ).

Ï�z� Wxi× Y � A ¥k��CX, ¤± X × Y �� A ¥k��CX. d A À�

�?¿5=� X × Y �;5. �

5 3.4.1 Tychonoff ½n�±?�Úäó, ?¿õ�;�m�ÈÿÀE´;�. ØLù�(Ø�y²´'�(J�, �©ò¬{�0�. �

½Â 3.4.4 � X ´ ÿ À � m, C ´ X � f 8 x, e é C ¥ ? Û k � f x{C1, C2, · · · , Cn}§Ù�8 C1 ∩ C2 ∩ · · · ∩ Cn o��, K¡ C ÷vk��^�.

½n 3.4.2 (;5�48�K) � X ´ÿÀ�m, K±e^��d:

(1) X ´;8;

(2) X ¥?�÷vk��^��48x C Ñk⋂

C∈CC 6= ∅;

(3) z�÷vk��^��f8x A , Ù��4��⋂

A∈AA 6= ∅.

y² (1)=⇒ (2) - A = {U | U = X − C,C ∈ C }, K A ´m8x. d⋃U∈A

U = X −⋂

C∈C

C, (3-1)

be C Ø÷v⋂

C∈CC 6= ∅, @o

⋃U∈A

U = X. d X �;5, �3 U1, U2, · · · , Un ∈ A , ¦�n⋃

i=1Ui = X. eP Ck = X − Uk ∈ C , @o�âª (3-1) �

n⋂k=1

Ck = ∅. ù� C �k��^�g

ñ! Ïd⋂

C∈CC 6= ∅.

- 37 -


(2)=⇒ (1) ^�y{. be X �;�, K�3mCX A = {Uα | α ∈ I}, ¦� A ¥k�fxÑØUCX X. - C = {C | C = X − U,U ∈ A }. éu C ¥?¿k�fx C1, C2, · · · , Cn ∈ C

C1 ∩ C2 ∩ · · · ∩ Cn = X − (U1 ∪ U2 ∪ · · · ∪ Un) 6= ∅

� C ÷vk��^�, d^� (2),⋂

C∈CC 6= ∅, ùL²

⋃U∈A

U 6= X, ù� A �À�gñ!

(2)=⇒ (3) � A ´÷vk��^��f8x, - C = {A | A ∈ A }. éu C ¥�?¿k�

�� A1, · · · , An, db�^��

A1 ∩A2 ∩ · · · ∩An ⊇ A1 ∩A2 ∩ · · · ∩An 6= ∅,

Ï C ÷vk��^�. d^� (2),⋂

A∈AA 6= ∅.

(3)=⇒ (2) ù´²��. �

íØ 3.4.2 � X ´;�m, �48@

C = {Cn | Cn ⊇ Cn+1, Cn 6= ∅, n = 1, 2, · · · , },

K C ÷vk��^�, l ∞⋂

n=1Cn 6= ∅.

AO/, é X = R1 þ�IOÿÀ, ?Û4«m@

C = {[an, bn] | [an, bn] ⊇ [an+1, bn+1], an < bn, n = 1, 2, · · · , }

Ñ÷v∞⋂

n=1[an, bn] 6= ∅.

½n 3.4.3 � X ´äkþ(.5��S8, K X 'uSÿÀ, Ùz�4«mÑ´;�.AO/, X = R1 3IOÿÀe, Ù4«mÑ´;�.

y² �½ a < b, �Ä4«m Y = [a, b]. � A ´ [a, b] ��f�mÿÀ��mCX,(ùp�f�mÿÀÚSÿÀ��) -

Σ = {y ∈ (a, b] | [a, y] U� A ¥k��CX}

·�©nÚ5y² Y ´;��.

(1) Äky² Σ ��.

e a k;�� a′, @o�±é� U, V ∈ A ¦� a ∈ U, a′ ∈ V . d� [a, a′] �±� U, V

CX. Ïd a′ ∈ Σ.

8� a Ã;��. d��3 A ¥�m8 U , ¦� a ∈ U . Ï� U ´m8, ¤±�3 (a, b]¥�� c0 ¦� [a, c0) ⊆ U . � y ∈ [a, c0),@o [x, y] � U CX, = y ∈ Σ.

(2) - c = supΣ, K a < c ≤ b. ·�5y² c ∈ Σ.

� V ∈ A ÷v c ∈ V . Ï� V ´m8, ¤±�3 d ∈ [a, c) ¦� (d, c] ⊆ V . b� c 6∈ Σ,K (d, c] ∩ Σ 6= ∅, eØ, d �´ Σ �þ., � c �´þ(.gñ! �½ z ∈ (d, c] ∩ Σ.5¿� [z, c] ⊆ (d, c] ⊆ V , ¿�d z ��{, [a, z] U� A ¥k��CX. Ïd

[a, c] = [a, z] ∪ (z, c] �U�k��CX, ù� c ∈ Σ, �b�gñ! ¤± c ∈ Σ.

- 38 -


(3) ��·�y² c = b. b� c < b.

-

Σ′ = {y ∈ (c, b] | [c, y] U� A ¥k��CX}.

aq1�Ú�y², �� Σ′ 6= ∅, =�3 y ∈ (c, b], ¦� [c, y] U� A ¥k��CX.du [a, c] �U� A ¥k��CX, ¤± [a, y] = [a, c] ∪ (c, y] U� A ¥k��CX. Ïd y ∈ Σ. ùÒÚ c �À�gñ! � c = b.

�d, ·��¤y². �

íØ 3.4.3 �Ä Rn þ�IOÿÀ, Y ⊆ Rn, K±e^��d:

(1) Y ´;8;

(2) Y 3î¼Ýþe´k.48;

(3) Y 3²�Ýþe´k.48.

y² � d ´î¼Ýþ, ρ ´²�Ýþ, �âØ�ª

ρ(x, y) ≤ d(x, y) ≤√nρ(x, y)

´� Y 3 d ¥k.��=� Y 3 ρ ¥k.. ,��¡ d Ú ρ p��Ó�ÿÀ, �Ò´` Y 3d ¥´48��=� Y 3 ρ �´48. Ïd��y² (1) � (3) ´�d�=�.

(1)=⇒ (3) � Y ´;8. Ï� Rn ´ Hausdorff �, d·K 3.4.3 ��, Y ´48.�Äm8x

B = {Bρ(0,m) | m ∈ Z+},

§�¿�u Rn, �þã8x´ Rn ¥��mCX. qÏ� Y ´;8, df�m;5�K, �3B ¥k��,

Bρ(0,m1), Bρ(0,m2), · · · , Bρ(0,mr)

¦� Y ⊆r⋃

i=1Bρ(0,mi). - M = max

1≤i≤r{mi}, =� Y ⊆ Bρ(0,M) . u´éu?¿� x, y ∈ Y k

ρ(x, y) ≤ 2M ≤ +∞. Ïd Y 'uÝþ ρ ´k.48.

(3)=⇒ (1) � Y ´48, �'uÝþ ρ k.. � m = supx,y∈Y

ρ(x, y). ?� x0 ∈ Y , -

M = m+ ρ(x0, 0), K

Y ⊆ [−M,M ]n := [−M,M ]× · · · × [−M,M ]︸︷︷︸n

.

Ï� [−M,M ] ´;8, ¤±È�m [−M,M ]n �´;8. d·K 3.4.2, Y ´;8. �

~ 3.4.7 (1) �Äü ¥¡ Sn−1 = {x ∈ Rn | ||x|| = 1}. Ï� Sn−1 ⊆ Rn, �§3IOÿÀe´k.48, ¤± Sn−1 ´;8.

(2) ü ¥N Dn = {x ∈ Rn | ||x|| ≤ 1}. Ó��, Dn ⊆ Rn ´k.48, ¤± Dn ´;8. �

- 39 -


~ 3.4.8 (1) �Ä8Ü Y1 = {(x, y) | y = 1x , 0 < x ≤ 1} ⊆ R2. Y1 ´ R2 ¥�48, �

Y1 Ã., Ïd Y1 Ø´;��.

(2) �Ä8Ü Y2 = {(x, y) | y = sin 1x , 0 < x ≤ 1} ⊆ R2. Y2 ´k.�, �§Ø´ R2 ¥�48,

Ïd Y2 �Ø´;8.

½n 3.4.4 � X ´;� Hausdorff �m. e X ¥�z�:Ñ´ X �à:, �Ò´`

X = X ′, @o X Ø�ê.

y² � x ∈ X,U ´ X ¥��m8. ·�©nÚy² X Ø�ê.

(1) ky², �3 y ∈ U ¦� x 6= y. e x 6∈ U , K?� y ∈ U =�; e x ∈ U , Ï� x ´ X �à:, Ïd�¹ x �?�m8� X ��kÉu x �:, �Ò´` U − {x} 6= ∅, =�?�

y ∈ U − {x}.

(2) 2y², �3��m8 V ⊆ U , ¦� x 6∈ V . � (1) ¥Éu x �: y. du X ´ Hausdorff�, ��3��m8 W1,W2 ¦�

x ∈W1, y ∈W2, � W1 ∩W2 = ∅

� V = W2 ∩ U . du W1 ∩ V = ∅, x ∈W1, �d4��K� x 6∈ V = V ∪ V ′.

(3) ��·�^�y{y² X Ø�ê. �y{, b� X ´�ê�,

X = {x1, x2, · · · }.

��m8 V1 ⊆ X, ¦� x1 6∈ V 1. aq/, ��m8 Vn ⊆ Vn−1 ¦� xn 6∈ Vn. ù�,·��48@,

V1 ⊇ V2 ⊇ · · · ⊇ Vn ⊇ · · ·

Ï� X ´;�m, díØ 3.4.2,∞⋂

n=1Vn. ·�� x ∈

∞⋂n=1

Vn 6= ∅, Ï� x ∈ X, X ´�ê

�, Ïd�3,� k, ¦� x = xk. ¤± xk ∈∞⋂

n=1Vn ⊆ Vk. �,, ù�«m@�À�gñ!

Ïd X Ø�ê. �

íØ 3.4.4 R1 ¥�4«m´Ø�ê8.

3.5 4�:;�S�;

½Â 3.5.1 X J ÿ À � m X ¥ � ? � Ã ¡ f 8 Ñ k à :, K ¡ X ´ 4 � : ;� ½Frechet ;, k��¡ X äk Bolzano-Weierstrass 5�.

·K 3.5.1 e X ´;8, K X �4�:;.

y² � Y ⊆ X ´Ã¡f8. �y, b� Y Ãà:, K Y = Y , Ïd Y ´48. qÏ� X

´;8, d·K 3.4.2 � Y ´ X �;f8.

- 40 -


?� x ∈ Y . Ï� x Ø´ Y �à:, ¤±�3 X ¥m8 Ux, ¦�

x ∈ Ux, Ux ∩ (Y − {x}) = ∅

w,, 8x A = {Ux}x∈Y ´ Y ��mCX, � Ux ∩ Y = {x}. d Y �;5, �3

Ux1 , Ux2 , · · · , Uxn∈ A

CX Y . Ï

Y = Y ∩ (n⋃

i=1

Uxi) =

n⋃i=1

(Y ∩ Uxi) = {x1, x2, · · · , xn}.

� Y ´Ã�:8, gñ!

5 3.5.1 �3ÿÀ�m, §�4�:;, �Ø´;��. Ù¥�a~f�ûS8k'. d

?Ø2Ðm. k,��Öö�±ëw [Ma87]. �

·��Ñ4�:;�¿Ø;��ÿÀ�m�,�a.~f.

~ 3.5.1 3��ê8 N þÚ\XeÿÀÄ

{{1, 2}, {3, 4}, · · · , {2n− 1, 2n}, · · · }.

§)¤ N þ��ÿÀ T .

·�y²§´4�:;�. ¯¢þ, §�?Ûf8Ñkà:. �d, ·��Iy²z�ü

:8Ñkà:. éu?¿�ü:8 {2n}, �Ä: 2n − 1 �?�� U , dÿÀÄ��E��,2n− 1 ∈ {2n− 1, 2n} ⊆ U . ùL²

(U − {2n− 1}) ∩ {2n} = {2n} 6= ∅.

Ïd 2n− 1 ´ {2n} �à:. Ón 2n ´ {2n− 1} �à:.

,��¡, �Ä N ¥��qm8

A = {{1, 2}, {3, 4}, · · · , {2n− 1, 2n}, · · · }

w� A ´ N ��qCX, � A ¥Ø�3k��m8CX N. � (N, τ) ¿Ø;�. �

½Â 3.5.2 � X ´ÿÀ�m, {xn}∞n=1 ´ X �:�, x0 ∈ X. XJé x0 �?Û�� U ,�3 N > 0, � n ≥ N �, ok xn ∈ U , K¡ {xn}∞n=1 Âñu x0.

·K 3.5.2 � (X, d) ´Ýþ�m, {xn}∞n=1 ´ X �:�, K±e^��d.

(1) {xn}∞n=1 Âñu x0;

(2) ∀ n > 0, ∃ N > 0, ¦�� k > N �, ok xk ∈ Bd(x0,1n).

^�¤á�, é?Û ε > 0, �3 N > 0, � n,m > N �, ok d(xn, xm) < ε.

y² (1)=⇒(2) 5g½Â, y²´²��.

(2)=⇒(1) � U ´ x0 ��, K�3 Bd(x0, ε) ⊆ U . �¿©�� n ¦� 1n < ε. Ï

Bd

(x0,

1n

)⊆ Bd(x0, ε) ⊆ U

- 41 -


d^�, �3 N > 0, ¦�� k > N �, xk ∈ Bd(x0,1n) ⊆ U .

8b�^�¤á. ?� ε > 0. ·��é��ê h, ¦� 1/h < ε/2. d^� (2), �3 N > 0,� n,m > N �ok

xn, xm ∈ Bd

(x0,

1h

)⊆ Bd(x0, ε/2).

dn�Ø�ª=� d(xn, xm) < ε. �

·K 3.5.3 e X ´ Hausdorff �, {xn}∞n=1 Âñ, KÂñ:´��.

y² eØ,, �� x0, y0 ´ü�ØÓ�Âñ:, @o©O�3�¹ x0, y0 �Ø��m8

U, V . d½Â, �3 N > 0, � k > N �, ok xk ∈ U, xk ∈ V . ù� U ∩ V = ∅ gñ! �

½Â 3.5.3 e X ¥?ÛS�ÑkÂñfS�, K¡ X S�;½{¡�;.

·��y²Xe�(Ø.

½n 3.5.1 � (X, d) ´Ýþ�m, K±e^��d:

(1) X ´;8;

(2) X 4�:;;

(3) X S�;.

�y²d(Ø, ·�I��O�ó�.

Ún 3.5.1 � X S�;�ÿÀ�m, ε ´�½�ê, @o�3k�� ε-¥

Bd(x1, ε), · · · , Bd(xn, ε)

CX X, = X =n⋃

i=1Bd(xi, ε).

y² �y{. b�é, ε > 0, X ØUdk�� ε-¥CX. 8�ES� {xn}∞n=1 Xe: ?

� x1 ∈ X. Ï� Bd(x1, ε) 6= X, ¤±�3 x2 ∈ X −Bd(x1, ε). �gaí, �

xn ∈ X −n−1⋃i=1

Bd(xi, ε).

l d(xn, xi) ≥ ε, i = 1, 2, · · · , n− 1.

ùL² {xn}∞n=1 Ø¹Âñf�. ù� X ´S�;gñ! �

Ún 3.5.2 � (X, d) ´Ýþ�m. X ´4�:;�, @o X �;.

y² � {xn}∞n=1 ´ X ¥�:S�, - Y =∞⋃

n=1{xn}. e Y �k�8, KdÄT�n, �

3Ã¡� xn Ü, Ï w,�]ÑÂñfS�.

Ø�� Y �Ã�8. Ï� X 4�:;, ¤± Y kà: x0. 5¿ Bd(x0,1n) ∩ Y ¹kÃ�õ

��. eØ,,� n ¿©�±�, Bd(x0,1n) ∩ Y = ∅, � x0 �½Âgñ! � n1 > 0 ¦� xn1 ∈

Bd(x0, 1), 2� n2 > n1 ¦� xn2 ∈ Bd(x0,12), ...... �g� nk > nk−1 ¦� xnk

∈ Bd(x0,1k ). Ï

dd·K 3.5.2 � {xnk}∞k=1 Âñu x0. �

- 42 -


Ún 3.5.3 (V��êÚn) � (X, d) ´Ýþ�m, A ´ X �mCX. e X S�;, K�

3 δ > 0, ¦� X ¥z��»�u δ �f87�¹3 A ¥�,��¥. ùp� δ ¡�V��ê.

y² �y{. b�Ø�3ù�� δ, =é?¿� δ > 0, Ñ�3 X ¥�»�u δ �f8,§Ø¹3 A ¥�?Û��¥.

éu?¿� n > 0, �f8 Cn ¦� Cn ��» d(Cn) < 1n � Cn Ø¹u A ¥?Û��S.

� xn ∈ Cn, ·�äóù�� {xn} ÃÂñf�. eØ,, �é�f� {xnk}∞n=1 Âñu x0, K x0

¹u A ¥,�� U S.

Ï� U ´m8, ¤±�3 ε > 0, ¦� Bd(x0, ε) ⊆ U . �¿©�� k, ¦ xnk∈ Bd(x0,

ε2) �

1nk< ε

2 . é?Û x ∈ Bd(xnk, 1

nk), dn�Ø�ª,

d(x, x0) ≤ d(x, xnk) + d(xnk

, x0) <1nk

+ε

2< ε,

Ï x ∈ Bd(x0, ε), l Bd(xnk, 1

nk) ⊆ B(x0, ε). é?Û y ∈ Cnk

, du d(y, xnk) < 1

nk, �

y ∈ Bd(xnk, 1

nk). ¤±

Cnk⊆ Bd(xnk

,1nk

) ⊆ B(x0, ε) ⊆ U.

ù� Cnk�À�gñ! �

½n 3.5.1 �y² (1)=⇒(2)=⇒(3)®y. y3y² (3)=⇒(1). � A ´ X �mCX. Ï� X

´S�;�, ¤± A kV��ê δ. � ε = δ/3. dÚn 3.5.1 �3 X � ε−¥k�CX, z�¥

�» d ≤ 2δ3 . dÚn 3.5.3 þãz�¥Ñ¹u A ¥,��¥. Ïd A k��k�fCX. 2

3.6 ëYN�

3ù�!�c, ·��´��ÿÀ�m�5�. ly3m©, ·��?Øü�ÿÀ�m

X,Y �m�'X. XÛïÄü�ÿÀ�m�'XQ? �~��{Ò´ïá,aÜ·�N�, r

üöéXå5. ù«g�@3p��ê¥Ò®Ñy. ·�ïÄü��þ�m��{�´ïá¦

��m��5N�. aq��{�Ñy3C�ê¥��aÓ�N�¥.

3.6.1 ëYN��Ó�

±e� X,Y ´ü�ÿÀ�m.

½Â 3.6.1 � f : X → Y ´��N�. XJé Y ¥�z�m8 V , Ù��8

f−1(V )4= {x ∈ X | f(x) ∈ V }

�´ X ¥�m8, K¡ f ´ëYN�, ½{¡ f ´ëY�.

·K 3.6.1 ±e·K�d:

(1) N� f ´ëY�;

(2) é Y �ÿÀÄ¥�� B, f−1(B) ´ X ¥�m8.

- 43 -


y² (1)=⇒ (2) ù´²��.

(2)=⇒ (1) � V ´ Y ¥�m8, K V =⋃

α∈I

Bα. Ù¥ Bα ´ Y ¥ÿÀÄ��. u´

f−1(V ) =⋃α∈I

f−1(Bα)

´ X ¥�m8. �

~ 3.6.1 (1) (ðÓN�) � (X,T ) ´ÿÀ�m, �ÄðÓN�

Id : (X, T ) −→ (X,T ),x 7−→ x.

§w,´ëY�.

(2) (�¹N�) � (X,T ) ´ÿÀ�m, � Y ⊆ X, �Ä�¹N�

i : Y −→ X,

y 7−→ y.

§´ëY�. ù´Ï�, eb� V ´ X ¥�m8, @o

i−1 = {y ∈ Y | i(y) ∈ V } = Y ∩ V

´ Y ��f�mÿÀ�m8.

(3) (ÝKN�)

π1 : X × Y −→ X,

x× y 7−→ x,

π2 : X × Y −→ Y,

x× y 7−→ y.

§�Ñ´ëYN�. ± π1 �~, ?� X ¥�m8 U , §�� U × Y ´ X × Y ¥�m8, Ïd

π1 ´ëYN�.

(4) �½: a ∈ X, b ∈ Y , �ÄXe�i\N�

ib : X −→ X × Y,

x 7−→ x× b,

ja : Y −→ X × Y,

y 7−→ a× y.

d (2), ib, ja �ëY. �

·K 3.6.2 �N� f : R1 → R1 ´ R1 3IOÿÀe�N�. @of 3ÿÀ¿ÂeëY�

�=� f 3êÆ©Û¿ÂeëY.

y² (=⇒) ®� f 3ÿÀ¿ÂeëY.

∀ ε > 0, ∀x0 ∈ R1, - y0 = f(x0) 9

Vε = (y0 − ε, y0 + ε) = {y ∈ R1 | |y − y0| < ε}

Ï� f 3ÿÀ¿Âe´ëYN�, ¤± f−1(Vε) ´ R1 ¥�m8. qÏ� x0 ∈ f−1(Vε), ¤±�3

- 44 -


δ > 0, ¦�x0 ∈ (x0 − δ, x0 + δ) ⊆ f−1(Vε)

5¿,(x0 − δ, x0 + δ) = {x ∈ R1 | |x− x0| < δ}.

·��òþã?Ø#Qã�, éu?¿÷v |x− x0| < δ �: x, þk |f(x)− f(x0)| < ε. d=

êÆ©Û¥ëY5�½Â.

(⇐=) ®� f 3êÆ©Û¿ÂeëY.

?�m«m V = (a, b) ⊆ R1, �y f−1(V ) ´ R1 ¥�m8, =éu ∀x0 ∈ f−1(V ), �é�m8 (c, d), ¦� x0 ∈ (c, d) ⊆ f−1(V ).

- y0 = f(x0), w, y0 ∈ V . �¿©��ê ε, ¦� (y0 − ε, y0 + ε) ⊆ V . Ï� f 3êÆ©Û¿Âe´ëY�, ¤±�3 δ > 0, � |x− x0| < δ �, ok |f(x) − f(x0)| < ε. ù�·�Òé� x0 �� (x0 − δ, x0 + δ) ¦�

x0 ∈ (x0 − δ, x0 + δ) ⊆ f−1((y0 − ε, y0 + ε)) ⊆ f−1(V )

¤± f 3ÿÀ¿Âe´ëYN�. �

~ 3.6.2 �N� f : X −→ Y , Ù¥ X ´lÑÿÀ�m, K X �?¿f8Ñ´m8, Ï f 7ëY. �

~ 3.6.3 �ÄN�f : (R1, T1) −→ (R1

l ,T2)x 7−→ x,

ùp (R1,T1) ´ R1 þ�IOÿÀ, (R1l ,T2) ´ R1 þ�e�ÿÀ. Ï�e�ÿÀ¥�Ä��´Ã

X [a, b) (a < b) �/ª, f−1([a, b)) = [a, b) Ø´ R1 ¥�m8.Ïd f ØëY! ùL², =¦.�m��, �e��ØÓ�ÿÀ, @o��8Ü�ðÓN��7ëY.

�L5, �ÄN�g : (R1

l , T2) −→ (R1,T1)x 7−→ x.

éu (R1,T2) ¥�?¿m8 (a, b), d

g−1(a, b) = (a, b) =⋃

c∈(a,b)

[c, b),

��§´ (R1l ,T1) ¥�m8. Ïd g ´ëYN�. �

5 3.6.1 þ~L², éëYN� f : X → Y 5`,

(1) U ⊆ X ´m8ØU�y f(U) ´ Y ¥m8;

(2) f ´V�ØUíÑ_N� f−1 ´ëY�. �

·K 3.6.3 (ëY5�K) � f : X → Y , K±e^��d:

- 45 -


(1) f ´ëYN�.

(2) é X ¥?Ûf8 W , Ñk f(W ) ⊆ f(W ), d?� f(W ) ½Â�:

f(W )4= {y ∈ Y | �3 x ∈W,¦� y = f(x)}.

(3) é Y ¥?Û48 Z, f−1(Z) �´ X ¥�48.

y² (1) =⇒ (2) � x ∈ W , V ´ f(x) �?��. ·��y f(x) ∈ f(W ), = V ∩f(W ) 6= ∅, ½=�y f−1(V ) ∩W 6= ∅.

Ï � f ë Y, V ´ Y ¥ � m 8, ¤ ± f−1(V ) ´ X � m 8. q Ï � f(x) ∈ V , � x ∈f−1(V ), l f−1(V ) ´ x ��. du x ∈W , �d4��5�íÑ f−1(V ) ∩W 6= ∅.

(2) =⇒ (3) � Z ´ Y ¥�48, W = f−1(Z). ·��y W ´48, =y W = W , ½=

W ⊆W .

?� W ¥�: x, ·��Iy² x ∈W , = f(x) ∈ Z. 5¿� f(x) ∈ f(W ), �d (2) ��

f(x) ∈ f(W ) ⊆ f(W ) ⊆ Z = Z.

Ïd f(x) ∈ Z.

(3) =⇒ (1) � V ´ Y ¥�m8, K Z = Y −V ´ Y ¥�48.d^� (3) ��,f−1(Z) ´4

8. 5¿

f−1(V ) = f−1(Y − Z) = f−1(Y )− f−1(Z) = X − f−1(Z)

´ X ¥�m8. dëYN��½Â, f ´ëY�. �

e¡·��ÑÿÀÆ¥��Vg.

½Â 3.6.2 � f : X → Y ´lÿÀ�m X � Y �ëYN�. XJ�3��ëYN�

g : Y → X, ¦� f ◦ g = IdY 9 g ◦ f = IdX , K¡N� f ´l X � Y ��Ó�, ½{¡ X �Y Ó� ({P� X ∼= Y ). f �_N� g P� f−1.

5 3.6.2 f : X → Y Ó��du÷v±en�^�:

(1) f ´V�, �ó�, f Q´ü�q´÷�;

(2) é X ¥?Ûm8 U , f(U) �´ Y ¥�m8 (��u f−1 �ëY5);

(3) é Y ¥?Ûm8 V , f−1(V ) �´ X ¥�m8 (��u f �ëY5). �

5 3.6.3 X þ�ÿÀ5�Ò´��6u X þ�ÿÀ��5�.�ó�, XJ X � Y

Ó�, KT5��3 Y þ¤á. ·�®ÆL�;5!ëÏ5!Hausdorff 5�Ú�Ýþz�Ñ´ÿÀ5�. �

~ 3.6.4 �½¢ê a, b ∈ R1,Ù¥ a 6= 0, �ÄXeN�:

f : R1 −→ R1 g : R1 −→ R1

x 7−→ ax+ b y 7−→ y−ba

w,,N� f, g Ñ´ëY�,� g ´ f �_N�, ¤± f ´Ó�N�. �

- 46 -


~ 3.6.5 �ÄXeëYN�f : R+ −→ R+,

x 7−→ 1x ,

ùp R+ = (0,+∞) ´ R1 �f�m. f ´Ó�N�, � f−1 = f .

aq�, ·�kXeÓ�N�g : R∗ −→ R∗

x 7−→ 1x

ùp R∗ = R− {0} ´ R1 �f�m. �

~ 3.6.6 �ÄXeN�f : (−1, 1) −→ R1 g : R1 −→ (−1, 1)

x 7−→ x1−x2 y 7−→ y

1+√

1+4y2

N� f, g Ñ´ëYN�, � g ´ f �_N�, Ïd (−1, 1) ∼= R1. �

~ 3.6.7 �ÄXeN�:

f : [0, 1) −→ S1 := {(x, y) | x2 + y2 = 1}(⊆ R2)t 7−→ (cos 2πt, sin 2πt)

dêÆ©Û(Ø, f ëY� f−1 �3. � [0, 1) ¥m8 [0, 14), f(U) Ø´ S1 �m8, ù´Ï�Ø

�3 R2 ¥�¹ f(0) �m8 V , ¦� V ∩ S1 ⊆ f(U). Ïd f−1 ¿ØëY, l f ØÓ�. �

~ 3.6.8 �Ä~ 3.6.3 ¥�N�f : Rl −→ R1,

x 7−→ x,

ùp Rl ´e�ÿÀ. f ´ëYN�� f−1 �3. � f ØëY, � R1 � Rl ØÓ�. �

~ 3.6.9 �ÄN�f : (a, b) −→ (0, 1)

x 7−→ x− a

b− a

§´Ó�N�.�ó�, m«mÑÓ�uIOm«m (0, 1). aq/, [0, 1] ∼= [a, b]. � [0, 1] 6∼= R1,ù´Ï� [0, 1] 3IOÿÀe´;�, � R1 ¿Ø;. �

½Â 3.6.3 � f : X → Y ´ëY�ü�, Z = f(X). e f : X → f(X) ´Ó��, K¡ f

´ X → Y �ÿÀi\.

~ 3.6.10 N�f : [0, 1) −→ R2

t 7−→ (cos 2πt, sin 2πt)

Ø´i\�. ù´Ï�~ 3.6.7 ®L² f : [0, 1) → S1 Ø´Ó�. �

- 47 -


3.6.2 ëYN��E

~ 3.6.11 (~�¼ê) �ÄXeN�

f : X −→ Y

x 7−→ y0

ùp y0 ∈ Y ´�½:. ·�¡ù��¼ê´~�¼ê. §´ëY�. ù´Ï�?� Y ¥�m8

V ,Ù��

f−1(V ) =

{∅, e y0 6∈ VX, e y0 ∈ V

Ïd, f−1(V ) Ñ´ X ¥�m8. �

~ 3.6.12 (�¹N�) � Y ⊆ X,�Ä�¹N�:

i : Y ↪→ X

y 7−→ y

?� X ¥�m8 U , du i−1(U) = U ∩ Y ´ Y ¥�m8, ¤±�¹N� i ëY. �

~ 3.6.13 (N��EÜ) � f : X → Y 9 g : Y → Z Ñ´ëYN�, KN� g ◦ f : X → Z

�´ëY�. �

y² ?� Z ¥�m8 W , ·�k

(g ◦ f)−1(W ) = f−1(g−1(W ))

Ï� g ´ëY�, ¤± g−1(W ) ´ Y ¥�m8. qÏ� f ëY, ¤± f−1(g−1(W )) ´ X ¥�m8, = (g ◦ f)−1(W ) ´ X ¥�m8. d W �?¿5� g ◦ f ´ëY�. �

~ 3.6.14 (��N�) � f : X → Y ´ëYN�, A ´ X �f�m, K�p� f 3 A þ

��N�f |A : A −→ Y

a 7−→ f(a)

N� f |A �´ëY�. ù´Ï�?� Y ¥�m8 V , (f |A)−1(V ) = f−1(V ) ∩ A ´ A ¥�m8.·��±rN� f |A w�ü�N��EÜ f |A = f ◦ i, =Xe��ã

A

i @@@

@@@@

f |A // Y

X

f

>>}}}}}}}

ùpi : A→ X ´�¹N�. i � f Ñ´ëY�, �d~ 3.6.13 ��, f |A �´ëY�. �

~ 3.6.15 (��) � f : X → Y ´ëYN�,Z ´ Y �f�m, ÷v f(X) ⊆ Z, K�p�ÑXeN�:

f : X −→ Z,

x 7−→ f(x).

- 48 -


Ï�éu Z ¥?Û�m8 W , �é� Y ¥m8 V , ¦� W = Z ∩ V , f−1(W ) = f−1(V ) ´X ¥�m8, � f ëY. �

~ 3.6.16 (��*Ü) � f : X → Y ëY, Y ⊆ Z, K�p��*Ü�N�

f ′ : X −→ Z,

x 7−→ f(x).

·��±rN� f ′ w¤N� f Ú�¹N� i : Y → Z �EÜ f ′ = i ◦ f , =Xe��ã

X

f AAA

AAAA

f ′// Z

Y

i

??~~~~~~~

Ïd f ′ ´ëY�. �

~ 3.6.17 (Ó��) � f : X → Y Ó�, A ´ X �f�m, K f |A : A → f(A) ´Ó�.�ó�, f |A : A→ Y ´ÿÀi\.

·�5�yù�(Ø. �â5P 3.6.2, ·�I�©O�yXe^�:

(1) (f |A)−1 : f(A) −→ A �3. ùw,5gu f ´��N��b�^�;

(2) f |A : A → f(A) ´ëY�. 5¿� f |A 5guEÜN� (f ◦ iA) : A iA−→ Xf−→ Y , ¿òT

EÜN�� f(A) þ��, ùp,iA L«�¹N�. Ïd§´ëY�.

(3) (f |A)−1 : f(A) −→ A ´ëY�. ù´Ï�§´ f−1 ÏLk�3½Â�Ú��þ��.

Ïd,f |A ´Ó�N�. �

~ 3.6.18 (Ó�'X) ·�½ÂÿÀ�m�m�'X

X ∼= Y ⇐⇒ �3Ó�N� f : X → Y.

·�5�yþã�'X ∼= ´�d'X.

(1) g�5: Id : X −→ X ´Ó�N�, � X ∼= X.

(2) é¡5: e f : X −→ Y Ó�N�, KdÓ�N��½Â��, f−1 : Y −→ X w,�´Ó�N�. ùÒL² X ∼= Y %¹X Y ∼= X.

(3) D45: e X ∼= Y, Y ∼= Z, ·�5y² X ∼= Z. � f : X −→ Y ±9 g : Y −→ Z Ñ´Ó�N�. ·��Ly g ◦ f : X −→ Z ´Ó�N�=�. Äk, _N� (g ◦ f)−1 = f−1 ◦ g−1

�3. Ùg, d g � f �ëY5�� g ◦ f ëY. aq�� f−1 ◦ g−1 �ëY5.

Ïd g ◦ f ´Ó�N�, l X ∼= Z. �

~ 3.6.19 ~ 3.6.9 L², ?Ûm«m (a, b) Ñ� (0, 1) Ó�. ~ 3.6.6 L² R1 ∼= (−1, 1).Ï , ·�k R1 ∼= (−1, 1) ∼= (0, 1) ∼= (a, b). dD45�, R1 ∼= (a, b). 3ÿÀÆ�ÆS, ïÄm«mÚïÄ��vk�O, Ï�§�´Ó��. �

- 49 -


·K 3.6.4 (ëY¼êÛÜ�K) � f : X → Y ´l X � Y �N�, K±e^��d:

(1) f ëY;

(2) (ÛÜL«) é X �?¿mCX {Uα}α∈I , Ñk f |UαëY.

(3) (Å:ëY) éu X ¥?¿�: x, ±9 f(x) �z�� V , Ñ�3 x �� U , ¦�

f(U) ⊆ V .

y² (1) =⇒ (2) 5g½Â��;

(2) =⇒ (1) � V ⊆ Y ´m8, Ï�

(f |Uα)−1(V ) = f−1(V ) ∩ Uα

�þª�>´ X ¥�m8, ¤±d

f−1(V ) =⋃α∈I

(f−1(V ) ∩ Uα) =⋃α∈I

(f |Uα)−1(V )

�� f−1(V ) ´ X ¥�m8. d V À��?¿5, �� f ´ëYN�.

(1) =⇒ (3) ®� f ëY. ?� x ∈ X 9 f(x) �� V . Äk, x ∈ f−1(V ). Ï� f ëY, �f−1(V ) ´ x ��. Ï �3 X ¥�m8 U , ¦� x ∈ U ⊆ f−1(V ), � f(U) ⊆ V . ¯¢þ, ·

��±�� U = f−1(V ).

(3) =⇒ (1) � V ´ Y �m8, ?� x ∈ f−1(V ). ·�k f(x) ∈ V . d (3) �b�^�, �

3 x �� Ux,¦� f(Ux) ⊆ V , = x ∈ Ux ⊆ f−1(V ). Ïd

f−1(V ) =⋃

x∈X

Ux

´ X ¥�m8. �

·K 3.6.5 (Ê�Ún) � X = A ∪ B, ùp A,B Ñ´48. f : A → Y 9 g : B → Y Ñ

´ëYN�, �÷v f |A∩B = g|A∩B, K�3ëYN� h : X → Y , ¦�

h(x) =

{f(x), x ∈ A,g(x), x ∈ B.

y² � C ´ Y ¥�48. ·�k

h−1(C) = f−1(C) ∪ g−1(C).

Ï� f ëY, ¤± f−1(C) ´ A ¥�48. qÏ� A ´ X ¥�48, ¤± f−1(C) �´ X ¥�48. Ó��, g−1(C) �´ X ¥�48. ùÒíÑ h−1(C) ´ X ¥�48. �

5 3.6.4 3þã·K¥, eò A,B U� X ¥�m8, (Ø�é. d�, §Ò´ëY¼êÛÜL«�A~. �

~ 3.6.20 (1) �Äýé�¼ê

h : R1 → R1, x→ |x|.

-

A = (−∞, 0], B = [0,+∞).

- 50 -


w� A ∩B = {0}. h �±w¤Xeü�¼ê�ÊÜ.

f : A −→ R1, g : B −→ R1,

x 7−→ −x, x 7−→ x.

ù´Ï� f(0) = g(0) = 0, ¿� A,B Ñ´48, Ï ÷vÊ�Ún�^�. ù�,N� h ´ëY�.

(2) �ÄN�

f(x) =

{−1, x < 0,1, x ≥ 0.

§Ø´ëYN�. � A = (−∞, 0), B = [0,∞). d� B ´48, A ∩ B = ∅, f |A, f |B Ñ´ëY�. ��Ø÷vÊ�Ún^��Ò´ A �48. �

~ 3.6.21 (�I¼ê) �ÄXeN�

f : A −→ X × Y

a 7−→ (f1(a), f2(a))

Ù¥ f1 : A −→ X, f2 : A −→ Y . ±e^�*d�d:

(1) f ëY;

(2) f1, f2 ëY.

d�·�¡ f1(a), f2(a) ´ f ��I¼ê.

·�5�yù�(Ø. Äk£��e~ 3.6.1 ¥�ÝKN�:

π1 : X × Y −→ X,

x× y 7−→ x,

π2 : X × Y −→ Y,

x× y 7−→ y.

ùp π1, π2 Ñ´ëY�.

(1) =⇒ (2) : ·�r f1, f2 w�±eN��EÜ:

f1 = π1 ◦ f, f2 = π2 ◦ f.

d π1, π2, f 9 f �ëY5, íÑ f1, f2 �ÑëY.

(2) =⇒ (1) : �Ä X × Y ¥�ÿÀÄ U × V , ·��y² f−1(U × V ) ´m8. 5¿�

a ∈ f−1(U × V ) ⇐⇒ f(a) ∈ U × V ⇐⇒ f1(a) ∈ U, f2(a) ∈ V ⇐⇒ a ∈ f−11 (U) ∩ f−1

2 (V ),

Ïd·�kf−1(U × V ) = f−1

1 (U) ∩ f−12 (V )

Ï� f1, f2 Ñ´ëYN�, ¤± f−11 (U) Ú f−1

2 (V ) Ñ´ A ¥�m8, l f−11 (U) ∩ f−1

2 (V ) �

´ A ¥�m8, = f−1(U × V ) �´ A ¥�m8. ùÒíÑ f ´ëY�. �

- 51 -


~ 3.6.22 (ëê�) �ÄXeN�

f : [a, b] −→ R2

t 7−→ (x(t), y(t))

K f ëY��=� x(t), y(t) ëY. �

~ 3.6.23 �ÄN�:

F : R1 × R1 −→ R1

(x, y) 7−→ F (x, y) =

{xy

x2+y2 , (x, y) 6= (0, 0),

0, (x, y) = (0, 0).

F éuz�©þ5`´ëY�. � F ��¼ê��¿ØëY. ù´Ï� F ��3é��þ

F (x, x) =

{1/2, x 6= 0,0, x = 0

§Ø´ëY�, � F �ØëY. �

3.6.3 ëYN��ëÏ5

½n 3.6.1 � f : X −→ Y ´ëYN�� X ´ëÏ�, K f(X) �ëÏ. ùp

f(X)4= {y | �3 x ∈ X,¦� y = f(x)} = Imf.

y² ·�^�y{. b� f(X) ØëÏ�k©� f(X) = U ∪ V . � Y ¥��m8

U ′, V ′ ¦�U = f(X) ∩ U ′ V = f(X) ∩ V ′

w�

f−1(U) = f−1(U ′), f−1(V ) = f−1(V ′).

Ï� f ëY, � f−1(U) Ú f−1(V ) Ñ´ X ¥��m8. 5¿

f−1(U) ∩ f−1(V ) = ∅ f−1(U) ∪ f−1(V ) = f−1(U ∪ V ) = X,

Ïd§�´ X �©�, = X ØëÏ, gñ! Ïd f(X) ´ëÏ�. �

íØ 3.6.1 � f : X → Y ´Ó�N�, K X ëÏ��=� Y ëÏ, =ëÏ5´ÿÀ5�.

~ 3.6.24 ·�|^ëÏ5y²: m«m (0, 1) Ú4«m [0, 1] ØÓ�.

�y, b�kÓ� f : [0, 1] −→ (0, 1). - p = f(1), Z = (0, 1) − {p}. dÓ��

f |[0,1) : [0, 1) → Z 9½n 3.6.1, Z �ëÏ. � Z k©� Z = (0, p) ∪ (p, 1), gñ! ��(Ø.

aq�y (0, 1) 6∼= (0, 1] ±9 [0, 1] 6∼= (0, 1]. �

½n 3.6.2 (0�½n) � f : X −→ Y ´ëYN�� X ´ëÏ�, Y ´SÿÀ. ?�

a, b ∈ X, Ø�� f(a) < f(b), Ké?Û r ∈ (f(a), f(b)), Ñ�3 c ∈ X , ¦� f(c) = r.

y² �y, b� r 6∈ f(X). � A = (−∞, r) ∩ f(X), B = (r,+∞) ∩ f(X). u´

A ∪B = f(X), A ∩B = ∅

- 52 -


du f(a) ∈ A, f(b) ∈ B, � A,B ´ f(X) ¥��m8. ÏdùÒ�Ñ f(X) ��©�.,��¡, Ï� X ´ëÏ�, ¤± f(X) �ëÏ. ù�c¡?Øgñ. ùÒíÑ r ∈ f(X). �

íØ 3.6.2 (":½n) � f : [a, b] −→ R1 ëY, � f(a) < 0 < f(b), K�3 c ∈ (a, b), ¦

� f(c) = 0. AO/, e f(x) ´'u x �Ûgõ�ª, K f(x) = 0 k¢�.

½Â 3.6.4 (�´ëÏ) � X ´ÿÀ�m, x, y ∈ X. eN� γ : [a, b] −→ X ´ëY�, �÷v γ(a) = x, γ(b) = y, K¡ γ ´ X ¥d x � y ��^�´.

e X ¥z�é:ÑU^ X ¥�´ë�, K¡ X ´�´ëÏ.

·K 3.6.6 e X �´ëÏ, K X ´ëÏ�.

y² �y, b� X = A ∪ B ´��©�. � x ∈ A, y ∈ B. Ï� X �´ëÏ, ¤±�3d x � y ��´ γ : [0, 1] −→ X, ÷v γ(0) = x, γ(1) = y. Ï� γ ëY, ¤± γ([0, 1]) ëÏ. dÚ

n 3.3.1, �o γ([0, 1]) ⊆ A �o γ([0, 1]) ⊆ B. ù��´ γ �À�gñ. �

~ 3.6.25 � Dn = {x ∈ Rn | ||x|| ≤ 1}. §´�´ëÏ�. ù´Ï�?� x,y ∈ Dn, �EXeëYN�

f : [0, 1] −→ Rn

t 7−→ (1− t)x + ty

÷v f(0) = x, f(1) = y. qd

||f(t)|| ≤ (1− t)||x||+ t||y|| ≤ 1

��d f ½Â��´��á3 Dn ¥. Ïd Dn ´�´ëÏ�. aq/, ·��±y²m¥

B(x, ε) �´ëÏ. �

~ 3.6.26 Rn − {0} (n > 1) ´�´ëÏ�. ?� x,y ∈ Rn − {0}. � xy ´ë� x Ú y

��ã. ·�©±eü«�¹�Ä. e xy ØL�:, K xy ´�´; e xy L�:, K2�

z ∈ Rn − {0}, ¦� xz � yz ØL�:. ¤± xz + yz ´l x � y ��´. �

íØ 3.6.3 éu?¿� n > 1,Rn 6∼= R1.

y² �y, e Rn ∼= R1 (n > 1), K Rn−{0} ∼= R1−{0}. Ï� R1−{0} ØëÏ, Rn−{0}´ëÏ�, gñ! Ïd Rn 6∼= R1(n > 1). �

5 3.6.5 � m > n �, ·��k Rm 6∼= Rn. ØLy²¿Ø{ü. �

~ 3.6.27 y²: S2 6∼= S1.

·�^�y{. b��3Ó� f : S2 ∼= S1. ?� p1, p2 ∈ S1, �

q1 = f−1(p1), q2 = f−1(p2),

K S2 − {q1, q2} ∼= S1 − {p1, p2}. ,��¡, ¥¡þ?¿��ü�:�´ëÏ�, ��±þ��

ü�:7ØëÏ. Ïd��gñ! � S2 6∼= S1. �

- 53 -


Ún 3.6.1 � f : X → Y ´ëYN�, X �´ëÏ, K f(X) ´�´ëÏ�. AO/, e

X,Y Ó�, K X �´ëÏ��=� Y �´ëÏ.

y² ?� y1, y2 ∈ f(X), �é x1, x2 ∈ X, ¦� y1 = f(x1), y2 = f(x2). Ï� X �´ëÏ, ¤±�3 X ¥ë� x1, x2 ��´ γ : [0, 1] → X, ÷v γ(0) = x1, γ(1) = x2. ù�, ·��±

�E f(X) ¥ë� y1, y2 ��´ f ◦ γ : [0, 1] → f(X). �

~ 3.6.28 �Ä Sn−1 = {x ∈ Rn | ||x|| = 1} (n > 1). Sn−1 ´�´ëÏ�.

�ÄN�f : Rn − {0} −→ Sn−1

x 7−→ x||x||

ù´ëY÷�. d~ 3.6.26 ÚÚn 3.6.1, Sn−1 �´ëÏ. �

~ 3.6.29 �Ä X = [0, 1] × [0, 1] �i;SÿÀ. du X kþ(.�÷v0�½n, �§

´�5ëYÚ, ? §´ëÏ�. �´ X ¿Ø´�´ëÏ�. �

y² �y, � X ´�´ëÏ�, ?� p = 0 × 0, q = 1 × 1. � γ : [0, 1] −→ X ´ X ¥

l p � q ��^�´. γ(0) = 0 × 0, γ(1) = 1 × 1. d0�½n, éu?¿� x × y ∈ Z, Ñ�3t ∈ [0, 1], ¦� γ(t) = x× y. ?� x ∈ [0, 1], Ux = γ−1(x× (0, 1)) ´ [0, 1] ¥�m8. éuØÓ�x ∈ (0, 1), Ux pØ��. � qx ∈ Ux ∩Q. Ï� qx pØ�Ó, ¤±N�

ϕ : I −→ {qx}x∈(0,1)

x 7−→ qx

´��éA. ,��¡, «m [0, 1] ´Ø�ê�, {qx}x∈(0,1) ´�ê�, gñ! �

~ 3.6.30 (Î�m) � K = { 1n | n ∈ Z+}, ·�½Â

C = (K × [0, 1]) ∪ (0× [0, 1]) ∪ ([0, 1]× 0)

·�¡�m C �Î�m. òl�m C ¥íØ��ã 0× (0, 1) ��m C ′ = C − 0× (0, 1)¡�">Î�m. ·�kXe(Ø:

(1) C ´�´ëÏ�;

(2) C ′ ´ëÏ�;

(3) C ′ �´ØëÏ. �

y² (1) C ��´ëÏ5´w,�;

(2) - A = ([0, 1]× 0) ∪ (K × [0, 1]), w, A ´ëÏ�, 2�â A ⊆ C ′ ⊆ A ±9ëÏf8�4��, C ′ ½ëÏ.

(3) � p = 0 × 1, �y�3N� f : [0, 1] −→ C ′ ´ C ′ ¥l p m©��^�´. XJf−1({p}) Q´m8q´48, @o�âëÏ5�±íÑ f−1({p}) = [0, 1]. = C ′ ¥Ø�3ë� p

Ú A ¥�:��´. Ïd·��8I=z�y² f−1({p}) Q´m8q´48.

- 54 -


du {p} ´ C ′ ¥�48, f ëY, ¤± f−1({p}) ´ [0, 1] ¥�48. Ïd·��Iy²f−1({p}) ´m8. y3� p 3 R2 ¥�� V ¦ÙØ� X ¶��. ?¿�½ f−1({p}) ¥�:x0, ·��8I´y²�3 x0 �� U ¦� U ⊆ f−1({p}). ùÒíÑ f−1({p}) ´m8.

·�� x0 ��Ä�� U , ¦� f(U) ⊂ V . e¡y²ù�� U =��é�m8,= U ⊆ f−1({p}). 5¿� U ´ [0, 1] þ'uSÿÀ��Ä��, §´ëÏ�, Ïd f(U)´ëÏ�. ¤± f(U) Ø¹?ÛÉu p �:. ù´Ï�éu C ′ ¥?¿3 V ¥�Éu p �:q = (1/n)× t0, À� r ¦� 1/n+ 1 < r < 1/n. @o�Ä R2 ¥ü�Ø��mf8:

(−∞, r)× R1 Ú (r,+∞)× R1

Ï� f(U) �¹3 D′ ¥�Ø� X ¶��, §�Ø�� x = r ��. Ïd§�¹3þãü�8Ü�¿¥. du f(U) ëÏ��¹1��8Ü¥�: p, ¤±§ÒØU�¹1��8Ü�:q, ùÒ%¹X f(U) = {p}. y.. �

~ 3.6.31 � S L«²¡þ�e�f8:

S = {x× sin(1/x) | 0 < x ≤ 1}

du S ´ëÏ8 (0, 1] 3ëYN�e��, Ïd S ½ëÏ. aquþ~�y², S Ø´�´

ëÏ. �

5 3.6.6 þãn~Ó�L², ��mëÏØ�½�´ëÏ, =·K 3.6.6 �_·KØ�

½¤á. �

½Â 3.6.5 ·�½Â X þ�Xe'X:

x ∼ y4⇐⇒ �3l x � y ��´

þã ∼ ´�d'X (�ÖögC�y). X 3 “ ∼ ” e��da¡��´ëÏ©|.

~ 3.6.32 �Ä¢�� R1 �f�m Y = [−1, 1] − {0}. d� Y ��´ëÏ©|ÚëÏ©|�Ó, ©O´ [−1, 0) Ú (0, 1]. �

~ 3.6.33 ~ 3.6.30 �">Î�m�k��ëÏ©|, �kü��´ëÏ©|. �

~ 3.6.34 (1) � X = R2 −A, ùp A ´²¡¥��õ�êf8. y²: X ´�´ëÏ�.

�y{. b�k x,y ∈ X, ¦�3 R2 ¥?Ûë� x,y ��´Ñ¹k A ¥�:.

·�é�^ØL x,y �� L. é?�: z ∈ L, ·��Eë� x,y �ò��´ xz +zy.éØÓ� z, ù�´Øà:�pØ��.

db�^�, éz� z ∈ L, �3þã�´¥é A ¥�: pz. Ã pz pØ�Ó. ù�·�kü

N� ψ : L → {pz}z∈L. ÃØXÛ, L ´Ø�ê�, {pz}z∈L ´�õ�ê�, gñ! Ïd7�3�

^ò��´��¹u X ¥.

(2) � Y = S2 −B, ùp B ´¥¡¥��õ�êf8. y²: Y ´�´ëÏ�.

e B ´k�:8, (Ø´w,�. Ø�� B = {pn}∞n=1 ´Ã�:8. �Ä¥4ÝK

pr : S2 − {p1} −→ R2.

- 55 -


§´��Ó�. ·��±��Ó��

S2 −B ∼= R2 − {pn}∞n=2.

(Üþ��K, á�(Ø. �

3.6.4 ëYN��;5

·K 3.6.7 � f : X → Y ëY, X ´;�m, K f(X) ½;. AO/, e f : X → Y ´Ó

�N�, K X ´;��=� Y ;, =;5´ÿÀ5�.

y² � Y ¥?�m8x A CX f(X), Kd f �ëY5��

A ′ = {f−1(V ) | V ∈ A }

´ X �mCX. Ï� X ´;�, ��3 V1, V2, · · · , Vn ∈ A , ¦�

f−1(V1) ∪ f−1(V2) ∪ · · · ∪ f−1(Vn) = X

¤± f(X) ⊆ V1 ∪ V2 · · · ∪ Vn. �âf�m�;5�K, f(X) ´;�. �

·K 3.6.8 � f : X → Y ëY, X ´S�;�, K f(X) �´S�;�.

y² � {yn}∞n=1 ´ f(X) ¥��G:�, K�3 x1, x2, · · · , xn, · · · ¦� f(xi) = yi. Ï�

X ´S�;�, ��3 X ¥�: x0, ±9Âñf� {xnk}∞k=1, ¦� lim

k→+∞xnk

= x0.

- f(x0) = y0. � U ´ y0 �?��, Kdu f ëY, � f−1(U) ´ x0 ��. dS�Âñ, �3 K > 0, ¦�éu?¿� k > K, Ñk xnk

∈ f−1(U). ù%¹X ynk∈ U . Ïd

{yn}∞n=1 �f� {ynk}∞k=1 Âñu y0. d {yn}∞n=1 À��?¿5�� f(X) S�;. �

5 3.6.7 Öö�U¬¯: eò·K 3.6.7 ¥� “;5” U� “4�:;”, §�ëY��½´4�:;�í? �Y´Ä½�. Öö�±ë�~ 3.5.1 gCéÑ�~. d, ëYN��¿Ø

�±�Ýþz, Hausdorff �ÿÀ5�. �

~ 3.6.35 £�~ 3.6.24. ·��±|^;55y² [0, 1) 6∼= [0, 1] ØÓ�. Ï� [0, 1] 3R1 ¥´;�, � (0, 1) Ø;. w,üöØÓ�.

Ón, ·��±y² [0, 1) 6∼= S1 (�~ 3.6.7). ù´Ï� [0, 1) Ø;, � S1 3IOÿÀe´k.48, Ïd´;�. ¤±üöØÓ�. �

~ 3.6.36 �Ä~ 3.6.28 ¥½Â�ëYN�:

f : Rn − {0} −→ Sn−1,

x 7−→ x||x|| .

f ´ëY÷�, Sn−1 ´;8, � Rn − {0} �;. �

~ 3.6.37 �ÄN�:

f : R1 × R1 −→ R1

(x, y) 7−→ xy

- 56 -


� A = {0} ⊆ R1. �Ä f−1(A) ��

f−1(A) = R1 × {0} ∪ {0} × R1

ùp f ëY, A ´;�, � f−1(A) �;. �

5 3.6.8 þãü~L²;8��Ø�½´;�. =·K 3.6.7 �_·KØ�½¤á. �

·K 3.6.9 � f : X → Y ´ëYV�. e X ´;�m, � Y ´ Hausdorff �, K f Ó�.

y² �â5P 3.6.2, ��y² f−1 ëY=�. qdëY5�K, ·��Iy²éu X ¥

�?�48 A, (f−1)−1(A) = f(A) ´ Y ¥�48=�.

Ï� X ´;�, A ⊆ X �´ X ¥�48. �â·K 3.4.2, A ´;�. �Ä f 3 A þ��

�. d��N��ëY5�� f|A : A −→ f(A) ´ëY�. d·K 3.6.7, f(A) ´ Y �;f8. Ï

� Y ´ Hausdorff �, d·K 3.4.3 �� f(A) = (f−1)−1(A) ´ Y ¥�4f8. Ïd f−1 ëY.�

½n 3.6.3 (��½n) � f : X → Y ´ëY�, Y ´äkSÿÀ�kS8, e X ´;�,K�é� a, b ∈ X ¦�éu?¿� x ∈ X, Ñk f(a) ≤ f(x) ≤ f(b).

y² Ï� f ëY � X ´;�, � f(X) ½;. �y f(X) k�� M , �� m. �y,Ø�b� f(X) Ã��. �E f(x) �XemCX

{(−∞, y) | y ∈ f(X)}

Ï� f(X) ´;�, ¤±�3 y1, y2, · · · , yn ∈ f(X) ¦� f(X) ⊆n⋃

i=1(−∞, yi). y3� M =

max{y1, · · · , yn}. K f(X) ⊆ (−∞,M). =éu?¿� x ∈ X, f(x) < M . ��â M �À�, �

3 i, ¦� yi = M . du yi ∈ f(X), ��3 x ∈ X ¦� f(x) = yi = M , gñ!

Ïd f(X) 7k��, Ón�y f(X) 7k��. �

íØ 3.6.4 (4«mþëY¼ê��½n) � f : [a, b] → R1 ëY, K f k�� (�) �.

~ 3.6.38 � f : S1 → R1 ëY, K f k��(�)�. ��íØ´, ¢ê�þ�ëY±Ï¼êk��(�)�. �

(Ü0�½n��½n, ·�kXek�(Ø.

·K 3.6.10 � f : X → Y ´ëYN�, Y ´äkSÿÀ��m, X ´ëÏ;�m, @o�8 f(X) ´ Y ¥�4«m.

y² d��½n±9 X �;5, �3 a, b ∈ X, ¦� f(X) ⊆ [f(a), f(b)]. qd0�

½n9 X �ëÏ5, é?Û r ∈ [f(a), f(b)], Ñ�3 c ∈ X ¦� f(c) = r. ùL² f(X) =[f(a), f(b)]. �

íØ 3.6.5 Rn ¥¥¡ Sn−1 þ�ëY¢�¼ê��8´ R1 �4«m. �ó�, � f :Sn−1 → R1 ´ëYN�, K f(X) ´4«m.

~ 3.6.39 � f : S2 → R1 ´ëYN�. y²: �3 t ∈ R1 ¦� f−1(t) ´Ø�ê8. ?�

Ú, 3 f(S2) ¥�õ�kü�ù��:, §��´�õ�ê8.

e f ´~�N�, K(Øw,. Ø�� f �~�. dþíØ, f(S2) = [m,M ] ´4«m. é?Û t ∈ (m,M), Ï� [m,M ] − {t} = [m, t) ∪ (t,M ] ØëÏ, ¤± S2 − f−1(t) �ØëÏ. d~3.6.34, ùÒíÑ f−1(t) ´Ø�ê8. �

- 57 -


3.6.5 ëYN��Ýþ

�!XÃAO(², þ%@ (X, dX), (Y, dY ) �Ýþ�m.

aq·K 3.6.2 �y², ·�kXe(Ø.

·K 3.6.11 (ëY¼ê� ε− δ �K) f : X → Y ëY�¿©7�^�´: éu?¿�

x ∈ X, ?¿� ε > 0, �3 δ > 0, ¦�� dX(x, y) < δ �k dY (f(x), f(y)) < ε.

·K 3.6.12 (ëY¼êS��K) � f : X → Y ´Ýþ�m�N�, K±e^��d:

(1) f ëY;

(2) éu X ¥�?�Âñu x0 �S� {xn}∞n=1, = limn→∞

xn = x0, Ñk limn→∞

f(xn) = f(x0).

3y²T·K�c·�k�ÑXeÚn.

Ún 3.6.2 (S�Ún) � A ⊆ X,x0 ∈ X, K±e^��d

(1) �3 A ¥S�Âñu x0;

(2) x0 ∈ A.

y² (1) =⇒ (2) � limn→∞

xn = x0, xn ∈ A. dÂñ5�½Â, x0 �?Û��¹k,�

xn ∈ A, = x0 �?�� A ��. Ïd x0 ∈ A.

(2) =⇒ (1) ®� x0 ∈ A, Kéu?¿� n > 0, Ñk Bd(x0,1n) ∩ A 6= ∅. ·��g�

xn ∈ Bd(x0,1n) ∩A. �â·K 3.5.2, lim

n→∞xn = x0. �

5 3.6.9 þã(Ø¥ (1)=⇒ (2) Ø�6uÝþ, é��ÿÀ�m�¤á. �

y3·�5y²·K 3.6.12

y² (1) =⇒ (2): � limn→∞

xn = x0. �y limn∞

f(xn) = f(x0). - V � f(x0) �?��.

= x0 ∈ f−1(V ). Ï� limn→∞

xn = x0, ¤±�3 N > 0, ¦�� n > N �, xn ∈ f−1(V ). =

f(xn) ∈ V . ¤± limn→∞

f(xn) = f(x0).

(2) =⇒ (1) �âëY5�K, �Iyéu?¿� A ⊆ X, Ñk f(A) ⊆ f(A). éu?¿�x0 ∈ A, �âS�Ún, �3 A ¥S� {xn}∞n=1 ¦� lim

n→∞xn = x0. Ïd {f(xn)}∞n=1 ⊆ f(A).

d (2) �� limn→∞

f(xn) = f(x0). 2g¦^S�Ún, ·�k f(x0) ∈ f(A). d x0 �?¿5,

f(A) ⊆ f(A). �

·K 3.6.13 � f : X → R1, g : X → R1 Ñ´ëY�. K f ± g ëY, f · g �ëY. e

g(X) ⊆ R1 − {0}, Kfg �ëY.

y² �ÄN�F : X → R1 × R1, x→ (f(x), g(x)).

d~ 3.6.21, F ´ëY�. 2�

G : R1 × R1 → R1, (a, b) → a+ b.

- 58 -


G �´ëY�, Ï G ◦ F : X → R1, (G ◦ F )(x) = f(x) + g(x)

´ëY¼ê. Ù¦�/aq�y. �

½Â 3.6.6 � fn : X → Y (n = 1, 2 · · · ) 9 f : X → Y ´N�. XJé?¿ ε > 0, �3

N > 0, ¦�� n > N �, é?Û x ∈ X, k

dY (fn(x), f(x)) < ε,

K¡N�S� {fn}∞n=1 ��Âñu f .

½n 3.6.4 (��4�½n) � fn : X → Y (n = 1, 2, · · · ) ´ëYN�. e {fn}∞n=1 ��Âñu f : X → Y , K f ½ëY.

y² � V ´ Y ¥m8. x0 � f−1(V ) ¥��:. ·��é� x0 �� U , ¦�f(U) ⊂ V .

- y0 = f(x0). kÀ� ε ¦� B(y0, ε) �¹3 V ¥. Ï� fn ��Âñu f , �éuÀ�� ε,�3 N > 0, ¦�éu?¿� n > N ±9?¿ x ∈ X Ñk d(fn(x), f(x)) < ε/4.

Ï� fN ëY, � f−1N (B(fN (x0), ε/2)) ´ X ¥�m8, � x0 ∈ f−1

N (B(fN (x0), ε/2)), K�

3 x0 �,�� U , ¦� fN (U) ⊆ B(fN (x0), ε/2).

ey f(U) ⊆ B(y0, ε), l f(U) ⊆ V . 5¿�� x ∈ U �, �¿©�� N Úþã� U ¦�

d(f(x), fN (x)) < ε/2 d(fN (x), fN (x0)) < ε/2 d(fN (x0), f(x0)) < ε/4

rùn�ªf�\, |^n�Ø�ª�� d(f(x), f(x0)) < ε. �

½Â 3.6.7 � f : X → Y , e?¿�½ ε > 0, �3 δ > 0, ¦�éu?¿� x1, x2 ∈ X, �� dX(x1, x2) < δ, Òk dY (f(x1), f(x2)) < ε, K¡ f ��ëY.

5 3.6.10 þã½Â¥� δ ��ûu ε, ¿Ø�6u x1, x2 �À�. �

½n 3.6.5 (��ëY5½n) � f : X → Y ´ëY�, X ´;�m, K f ��ëY.

y² �½ ε > 0, � Y ��± ε2 ��»�¥ B(y, ε

2) �¤�mCX. � A ´dù¥3 f e��¤|¤� X �mCX. dV��êÚn, �3 δ > 0, ¦� X ¥z��»�u δ �f87�¹3 A ¥�,��¥. y3� δ �CX A ��V��ê. XJ x1 Ú x2

´ X ¥÷v dX(x1, x2) < δ �ü�:, @où�ü:8 {x1, x2} ��»Ò�u δ, ¤±§��

{f(x1), f(x2)} 73,�¥ B(y, ε2) ¥. Ïd dY (f(x1), f(x2)) < ε. �

- 59 -

1oÙ :8ÿÀ (III): �\E|


4.1 �ê5ún

½Â 4.1.1 � X ´ÿÀ�m, x ∈ X. e�3 x ��êx B, ¦� x �z��¹ B ��, K¡ X 3 x ?k�êÄ.

e X �z�:�k�êÄ, K¡ X ÷v1��ê5ún(�¡ C1 ún). ÷v C1 ún�ÿ

À�m X �¡� C1 �m.

~ 4.1.1 ·��yÝþ�m (X, dX) ´ C1 �m. éu X ¥?¿� x, � x ��x��

B ={B

(x,

1n

)| n ∈ Z+

}.

éu x �?��o�¹þã�êx B ¥��, � X 3: x ?k�êÄ. d x �?¿5, X÷v1��ê5ún. AO/, lÑÿÀ´ C1 �m. �

5 4.1.1 (1) �êÄ B �À�Ø��. 'X3þ~¥, e�

B′ = {B(x, q) | q ∈ Q+},

K B′ �´ x ?��êÄ.

(2) � X 3 x ?k�êÄ, K�3 x ?��ê��Ä B = {Vn}∞n=1 ÷v

V1 ⊇ V2 ⊇ · · · ⊇ Vn ⊇ · · ·

ù´Ï�éu x �?¿�êÄ {Un}∞n=1, �- Vn = U1 ∩ U2 ∩ · · · ∩ Un, =�÷v^��ê�

�Ä {Vn}∞n=1. �

½n 4.1.1 � X ´ C1 �m, A ⊆ X, K

(1) x ∈ A ��=��3 A ¥:� {xn}∞n=1, ¦� limn→∞

xn = x.

(2) � f : X −→ Y ´N�, @o f ëY�¿©7�^�´: éuz�Âñu x �S�{xn}∞n=1, ÑkS� {f(xn)}∞n=1 Âñu f(x).

y² y²aqu X ´Ýþ�m��/. �Iò “¥” U��êÄ=�. �

½Â 4.1.2 e X k�êÿÀÄ, K¡§÷v1��ê5ún (q¡ C2 ún). ÷v C2 ú

n�ÿÀ�m X ¡� C2 �m.

5 4.1.2 (1) eÿÀ�m X ´ C2 �m, K§w,´ C1 �m.

(2) �3Ø÷v C2 ún�Ýþ�m. �

~ 4.1.2 � X = Rn, du Rn �3�êÿÀÄ

B = {(a1, b1)× (a2, b2)× · · · × (an, bn) | ai, bi ∈ Q}

� X ´ C2 �m. �

- 60 -


½n 4.1.2 � X ´ C1 �m (�A/, C2 �m), K

(1) � A ´ X �f�m, A ⊂ X, @o A �´ C1 �m (�A/, C2 �m). �ó�, �ê5ú

n�¢D�f�m;

(2) � Y �´ C1 �m (�A/, C2 �m), K X � Y �¦È�m X × Y �´ C1 �m (�A/, C2 �m).

(3�ÖögC�y.)

½Â 4.1.3 � X ´ÿÀ�m, A ⊆ X. e A = X, K¡ A 3 X ¥È�; e X k�ê�È�f8, K¡ X ´�©�, ½` X ´�©�m.

½n 4.1.3 � X ´ C2 �m, K

(1) (Lindelof ^�) X �?ÛmCXÑ�¹�êfCX;

(2) X ´�©�.

y² (1) � A ´ X �mCX, B = {Bn}∞n=1 ´ X ��êÄ. -

Σ = {n ∈ Z+ | �3 A ¥��¹ Bn}

Äk, ·�y² Σ ��. éu?¿� x ∈ X, �3 A ∈ A ¦� x ∈ A. dÿÀÄ�½Â, �3Bn ∈ B ¦� x ∈ Bn ⊆ A.

e¡·�y² A �¹�êfCX. éu?¿� n ∈ Σ, � An ∈ A ¦� Bn ⊆ An. P

A ′ = {An | n ∈ Σ}

ey A ′ ´�êfCX. Ï�eI8 Σ ´�ê�, � A ′ �ê. éu?¿� x ∈ X, dþ?Ø, �

3 Bn ∈ B, ¦� x ∈ Bn ⊆ An, � x ∈ An, d x �?¿5, A ′ CX X.

(2) � B = {Bn}∞n=1 ´ C2 �m��êÄ. 3z� Bn ¥��: xn (n = 1, 2, · · · ). P D ={xn}∞n=1. ·�5y² D ´ X ��êÈ�f8. d D ��{, �ê5´w,�. ey D = X.

éu?¿� x ∈ X, � B ´ x �?¿Ä��, K�3 n > 0, ¦� B = Bn. Ïd xn ∈ Bn∩D%¹X B ∩D 6= ∅. ùÒk x ∈ D. d x �?¿5� X = D. �

½Â 4.1.4 eÿÀ�m X ÷vþã½n (1) ¥�5�, =§�?ÛmCXÑ�¹�êfCX, ·�Ò¡ X ´ Lindelof �m. þ¡½näó C2 �mÑ´ Lindelof �m.

½n 4.1.4 � (X, d) ´Ýþ�m, K±e^��d

(1) X ´ C2 �m;

(2) X ´ Lindelof �m;

(3) X ´�©�m.

- 61 -


y² (1) =⇒ (2) 5g½n 4.1.3.

(2) =⇒ (3) �½ n > 0, �Äm8x {B(x, 1n) | x ∈ X}. §´ X �mCX. �d Lindelof ^

��, �3�êfCX, P� Bn. - B =∞⋃

n=1Bn. B ´�êCX. ·�3 B �z��¥�Ù

¥%, ù¥%�¤�8ÜP� D. ·��y D ´�êÈ�f8.

D ��ê5w,. eyÙÈ�. ?� x ∈ X ±9 x �?ÛÄ�� B(x, ε). é n > 1/ε±9 Bn ¥�¥ Bd(y, 1

n), ¦� x ∈ Bd(y, 1n). Ï� d(x, y) < 1

n < ε, ¤± y ∈ Bd(x, ε), l y ∈ Bd(x, ε) ∩D. ÏddÄ��?¿5�� x ∈ D. 2d x �?¿5=� X = D.

(3) =⇒ (1) ®� X ´�©Ýþ�m, A ´�êÈ�f8. �

B = {B(a,1n

) | a ∈ A,n ∈ Z+}

·�y²þã� B ´�êÄ, Ï X ´ C2 �m. B ��ê5´w,�. ey B ´ X �ÿÀ

Ä, =éu?¿�m8 U , �é� B ∈ B ¦� x ∈ B ⊆ U .

Äk, dÝþÿÀÄ�À�, �3 ε > 0, ¦� x ∈ Bd(x, ε) ⊆ U . éuþã� ε, ·�é��

ê n > 2/ε, Ï

x ∈ B(x,

1n

)⊆ B(x, ε) ⊆ U

Ùg, Ï� A = X. ¤± B(x, 1n) ∩ A 6= ∅. �� a ∈ B(x, 1

n) ∩ A. l d(x, a) < 1n < ε

2 , =

x ∈ B(a, 1n).

��, ·�y² B(a, 1n) ⊆ B(x, ε), l x ∈ B(x, 1

n) ⊆ U . ¯¢þ, é?¿� y ∈ B(a, 1n),

d d(y, a) < 1n 9n�Ø�ª

d(x, y) ≤ d(x, a) + d(a, y) <2n< ε

¤± y ∈ B(x, ε). �

íØ 4.1.1 � X ´;Ýþ�m, K X ´ C2 �m.

Ï~5`, �y�©5�'�y C2 5�N´�õ.

~ 4.1.3 î ¼ � m Rn ´ � © �, ' X · � � ± � Rn ¥ � È � f 8 A ={(x1, x2, · · · , xn) | xi ∈ Q}. Ïd§�´ C2 �. �

~ 4.1.4 (Hilbert �m) �Ä�5�m

Rω =

{(x1, x2, · · · ) | xi ∈ R,

∞∑n=1

x2n < +∞

}

½Â Rω þ�SÈ$�

〈{xn}, {yn}〉 =∞∑

n=1

xnyn

±9Ýþ

ρ({xn}, {yn}) =

√√√√ ∞∑n=1

(xn − yn)2

- 62 -


Ïd ρ p��5�m Rω þ�ÝþÿÀ. �

A = {(x1, x2, · · · ) ∈ Rω | xi ∈ Q �Øk��, Ù{�þ�"}

Ï� A ´ X = Rω ��êÈ�f8, ¤± Rω �©, ? ´ C2 �m. �

~ 4.1.5 �Ä¢ê8 R1 þ�e�ÿÀ R`.

(1) R` ´ C1 �m. Ï�éu?¿� x ∈ R`, {[x, x+ 1n) | n > 0} ´: x ?��êÄ.

(2) R` Ø´ C2 �m. � B ´ R` ��|ÿÀÄ. éu?¿� x ∈ R`, � Bx ∈ B ¦�

x ∈ Bx ∈ [x, x+ 1).

�,�: y ∈ R`, K By 6= Bx. eØ, x = inf Bx = inf By = y, gñ! ùÒ�Ñü�

ψ : Rl −→ {Bx}x∈R(⊆ B)x 7−→ Bx

Ï� R` Ø�ê, ¤± B Ø�ê. �

4.2 ©l5ún

� X ´ÿÀ�m. ÄkÚ\Xeo^©l5ún.

T1 ún éu X ¥?¿ü�ØÓ: x, y ∈ X, �3m8 U, V ¦�

x ∈ U y ∈ V, y 6∈ U, x 6∈ V

T2 ún (= Hausdorff 5�) éu?¿ü�ØÓ: x, y ∈ X, �3Ø��m8 U, V , ¦�

x ∈ U, y ∈ V.

T3 ún éu?¿� x ∈ X 9?¿Ø¹ x �48 F , �3Ø��m8 U, V ÷v

x ⊆ U, F ⊆ V.

T4 ún éu?¿ü�Ø��48 F1, F2, �3Ø��m8 U, V ÷v

F1 ⊆ U, F2 ⊆ V.

½Â 4.2.1 b� X ¥�ü:8´48. e X ÷v T3 (�A/, T4) ún, K¡ X ´�K

� (�A/, �5�), ½¡ X ´�K�m (�A/, �5�m).

5 4.2.1 e X �5, K X �K; e X �K, K X ÷v Hausdorff 5�. �Öög1�y

ùü�(Ø. �

- 63 -


·K 4.2.1 (�K5��55��K) � X ´ÿÀ�m, ü:8´48, @o

(1) X �K ⇐⇒ ?� x ∈ X ±9 x �?Û�� U , Ñ�3 x �� V , ¦� V ⊆ U .

(2) X �5 ⇐⇒ é?¿48 A ±9�¹ A �?Ûm8 U , �3�¹ A �� V , ¦� V ⊆ U .

y² (1) (=⇒) ®� X �K, x ∈ X, U ´ x ��. B = X − U ´ X ¥�48. d�K^��, �3m8 V,W ¦�

x ∈ V, B ⊆W, V ∩W = ∅

éu?¿� y ∈ B, W ´ y ��, � W ∩ V = ∅, ùÒíÑ y 6∈ V . � B ∩ V = ∅, u´ V ⊆ U .

(⇐=): � x ∈ X, B ´Ø¹ x �48. � U = X − B, d^�, �3 x �m�� V ¦�V ⊆ U . u´

x ∈ V, B ⊆ X − V , V ∩ (X − V ) = ∅.

Ïd X �K.

(2) aq�y. �

·K 4.2.2 � X ´�K�(Hausdorff �), K

(1) e A ⊆ X, @o A ��K(Hausdorff �);

(2) e Y ��K(Hausdorff �), KÈ�m X × Y �´�K�(Hausdorff �).

y² éu X ´ Hausdorff ��/, c©®k?Ø. y3�Ä X ´�K��/.

(1) � A ⊆ X. Ï� X �K, ¤± X ´ Hausdorff �. Ï� Hausdorff 5��¢D�f�m,� A �´ Hausdorff �. d·K 3.2.4 A ¥ü:8Ñ´48. � x ∈ A, B ´ A ¥Ø¹ x �4f8. �â4��úª

B = A ∩ ClX(B)

ùÒk x 6∈ ClX(B). d X ��K5, �3 X ¥m8 U, V ¦�

x ∈ U, ClX(B) ⊆ V, U ∩ V = ∅.

u´x ∈ U ∩A, B ⊆ V ∩A, (U ∩A) ∩ (V ∩A) = ∅

ùL² X �f�m A ´�K�.

(2) Ï� X,Y �K, �§�Ñ´ Hausdorff �. ¤± X × Y �´ Hausdorff �. l X × Y

¥z�ü:8Ñ´48. � x× y ∈ X × Y , W ´ x× y ��. �éÄ��÷v

x× y ∈ U × V ⊆W

Ï� x ∈ U � X ´�K�, �â·K 4.2.1 �3 X ¥�m8 U1 ¦�

x ∈ U1 ⊆ U1 ⊆ U

Ón��, �3 Y ¥m8 V1, ¦�y ∈ V1 ⊆ V1 ⊆ V

- 64 -


¤±

x× y ∈ U1 × V1 ⊆ U1 × V1 = U1 × V1 ⊆ U × V,

= x× y k�� U1 × V1, ¦� U1 × V1 ⊆W . d·K 4.2.1, X × Y �K. �

5 4.2.2 �5�m�f�m�7�5; �5�m�È�m��7�5. �

½n 4.2.1 � X ´ÿÀ�m, e X ÷v±e^��, K X �5.

(1) X ´Ýþ�m;

(2) X ´;� Hausdorff �m;

(3) (Lindelof ½n) X �K�´ C2 �m.

y² (1) � (X, d) ´Ýþ�m, A,B ´Ø��48. éu?¿� a ∈ A, �

B(a, εa) ∩B = ∅,

= B(a, ε) ⊆ X −B. ½Â

U =⋃a∈A

B(a,εa2

)aq/, ½Â

V =⋃b∈B

B(b,εb2

)ùp B(b, εb) ∩A = ∅.

8y U ∩ V = ∅ =�. �y, � z ∈ U ∩ V . K�3 a ∈ A, b ∈ B, ¦�

z ∈ B(a,εa2

) z ∈ B(b,εb2

)

l d(a, z) <

εa2

d(b, z) <εb2

dn�Ø�ªd(a, b) < d(a, z) + d(z, b) <

εa2

+εb2

=εa + εb

2

e εa < εb, K d(a, b) < εb, ùÒíÑ a ∈ B(b, εb), gñ! e εa ≥ εb, @o b ∈ B(a, εa), ½gñ!� U ∩ V = ∅.

(2) � x ´ X ¥�:, B ´ X ¥Ø¹ x �48. Ï� X ´;� Hausdorff �m, d·K3.4.2 ��, B ´;�. qdíØ 3.4.1 ��, ©O�3 x �� U Ú�¹ B �m8 V , ¦�U ∩ V = ∅. u´ X ´�K�.

� A Ú B ´ X ¥�Ø��48, éu?¿� a ∈ A, d�K5, �3m8 Ua, Va ¦�

a ∈ Ua, B ⊆ Va, Ua ∩ Va = ∅.

{Ua} ´ A �mCX. Ï� X ´ Hausdorff �� A ´ X ¥�48, ¤± A ´;f8. ��3k��m8 Ua1 , Ua2 , · · · , Uam

CX A. y3�

U = Ua1 ∪ Ua2 ∪ · · · ∪ UamV = Va1 ∩ Va2 ∩ · · · ∩ Vam

- 65 -


·�k A ⊆ U,B ⊆ V,U ∩ V = ∅. �55�y.

(3) � A,B ´Ø��48. X k�êÄ B. éu?¿� x ∈ A, Ï� A Ú B Ø��, ¤±

�3 x �� U ¦�x ∈ U, U ∩B = ∅

qÏ� X �K, �â·K 4.2.1, �3m8 V ¦�

x ∈ V, V ⊆ U

qÏ� B ´ÿÀÄ, ¤±�3 Bx ∈ B, ¦�

x ∈ Bx ⊆ V, Bx ⊆ U

l Bx ∩ B = ∅. Ï� B ´�êÄ, � {Bx}x∈A 3�KE�±�´ A ��êmCX, � Bx

Ø� B ��. Ón�� B ��êCX {B′y}y∈B, ¦� B′y Ø� A ��.

��Bå�, ò A �þãCXP� {Un}∞n=1, B �þãCXP� {Vn}∞n=1. -

U ′n = Un −n⋃

k=1

Vk, V ′n = Vn −n⋃

k=1

Uk, U ′ =∞⋃

n=1

U ′n, V ′ =∞⋃

n=1

V ′n

��y A ⊆ U ′, B ⊆ V ′, U ′ ∩ V ′ = ∅ ([!Ñ). �

·�Ø\y²/QãXe(Ø.

½n 4.2.2 ?ÛSÿÀ��5.

~ 4.2.1 � K = { 1n | n ∈ Z+} ⊆ R1.

B = {(a, b) | a < b, a, b ∈ R1} ∪ {(a, b)−K | a < b, a, b ∈ R1}

´ÿÀÄ. d§)¤�ÿÀ (R1,B) ´ Hausdorff �, �Ø´�K�! ù´Ï�e�

x = 0, A = K

Kdu A 3ÿÀ B e´ R1 ¥�48, � x 6∈ A. �´ x �?��o��¹ A ��, �Ø�K. �

:8ÿÀÆ¥��½n��=Xe��Ýþz½n.

½n 4.2.3 (Urysohn Ýþz½n) � X ´�K�, � X ´ C2 �m, K X �Ýþz.

4.3 Urysohn Ún� Tietze *Ü½n

½n 4.3.1 (Urysohn Ún) � X �5, A,B ´ X ¥ü�Ø��48. [a, b] ´ R1 ¥�

4«m, K�3ëYN� f : X −→ [a, b] ¦�

f(x) =

{a, e x ∈ Ab, e x ∈ B

½n 4.3.2 (Tietze *Ü½n) � X �5, A ´ X ¥�4f8. [a, b] ´ R1 �4«m. K

- 66 -


(1) ?ÛëYN� f : A −→ [a, b] �*Ü�ëYN� f : X −→ [a, b];

(2) ?ÛëYN� f : A −→ R1 �*Ü�ëYN� f : X −→ R1.

(Ï��, �ùÂ�U�)¤kSN, 3�8�Ö¿.)

4.4 Urysohn Ýþz½n

4.5 Tychonoff ½n

- 67 -

¢ Ú

¢ Ú

ε− δ �K, 58ε-¥, 15

Hausdorff �m�K, 26Hausdorff �m, 24Hilbert �m, 62

Lindelof �m, 61

Tietze *Ü½n, 66

Urysohn Ún, 66

Zariski ÿÀ, 6

�m«m, 10�¹N�, 44, 48

4�, 204��K, 2248, 54«m, 94��, 11>., 23IOÿÀ, 4

~�¼ê, 48

�8, 23�´, 53�´ëÏ©|, 551��ê5ún, 601��ê5ún, 60

Ýþ, 15Ýþ�m, 17ÝþÿÀ, 15é¡5, 15

©l5ún, 63©�, 28©�5�K, 29

CX, 34

+/Ún, 36

ðÓN�, 44

Ä��, 7ÈÿÀ, 11, 12

4�:, 234�:;, 40

0�½n, 52;�c�, 9;5�48�K, 37;�5, 34

à:, 23ål, 15

mCX, 34m8, 4m«m, 9m��, 11�Ýþz, 17�©, 61�êÄ, 60

V��ê, 43lÑÿÀ, 4

ëÏ©|, 33ëÏ5, 28ëY5�K, 45ëYN�, 43��, 22":½n, 53

#'¿d�, 2SÜ, 20

²�ÿÀ, 4²�Ýþ, 18

�S8, 9

- 69 -

¢ Ú

n�Ø�ª, 15

Âñ, 41Î�m, 54

ÝKN�, 12ÿÀ, 4ÿÀC�, 1ÿÀÄ, 7ÿÀ�m, 4ÿÀi\, 47

e�ÿÀ, 9��N�, 48�5ëYÚ, 32�ÿÀ, 12

S�;, 42SÿÀ, 10

��4�½n, 59��ëY5½n, 59��Âñ, 59

k��^�, 37

�½5, 15�5, 63�5�m, 63�K, 63�K�m, 63

f�m;5�K, 35f�mÿÀ, 13i;S'X, 9��½n, 57�I¼ê, 51

- 70 -

ph.d thesis model

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