Math 4181 Name:Dr. Franz RotheDecember 4, 201414FALL\4181_fall14h50.tex

Homework has to be turned in this handout.For extra space, use the back pages, or blank pages between.

The homework can be done in groups up to four studentsdue November 31th

5 Homework

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10 Problem 5.1. Let (X, I) and (Y,J ) be topological spaces andf : X 7 Y be any mapping. We know that a function is continuous if and only if(3) the preimage f1(B) X is closed for each closed set B Y .Prove that the function is continuous if and only if

(b) f1(C) f1(C) holds for each set C Y .Proof of (3) (b). The answer is left to the student.Proof of (b) (3). The answer is left to the student.

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Definition 1. A subset A X of a topological space X is called dense iff A = X.10 Problem 5.2. Convince yourself that a set A is dense if and only if either

(i) Any nonempty open set contains some point of A: O 6= A O 6= ; or(ii) the complement has empty interior: int(X \ A) = .

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10 Problem 5.3. Let the subset A X of a topological space X be open anddense, and the subset B X be dense. Prove that the intersection A B is dense.Proof. Let O 6= be any open nonempty set. Since the set A is dense, by item (i) ofthe last problem we conclude ? .

Moreover, this is an open set. Using item (i) once more, now for the dense set B,we conclude that the intersection ? is nonempty.

The entire paragraph implies, again via item (i) that the set A B is dense.10 Problem 5.4. Prove that the finite intersection of open dense sets is open

and dense.

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10 Problem 5.5. Let X, Y be topological spaces, A X be a dense subset, andf : X 7 Y a continuous function onto Y . Prove that f(A) is dense in Y .

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10 Problem 5.6. Let X be a topological space and A X be any subset. Forthis problem, I find it convenient to denote the interior by int(A) and the closure bycl(A). Prove the inclusions

int(A) int(cl(int(A))) int(cl(int(cl(A))))cl(int(cl(int(A))))

cl(int(cl(A))) cl(A)

10 Problem 5.7. Show that by applying the operations of interior and inclusionsuccessively to one set, no more than the six sets as in problem 5.6 can be obtained.Especially

cl(int(cl(int(cl(A))))) = cl(int(cl(A))) and

int(cl(int(cl(int(A))))) = int(cl(int(A)))

hold for all subsets A X.

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10 Problem 5.8. Show that by applying the operations of interior and inclusionsuccessively to an appropriate set S, indeed the six sets as in problem 5.6 turn out to beall distinct.

LetA :=

{(pq 22q, p

q+ 22q : 1 p < q and q 2

} (0, 1)

be an open dense set in the space [0, 1]. The Lebesgues measure of A is less than

|A|