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Jonathan Kim MATH 385 5/9/12

Review problems for Chapter 2.3

2.3.1 Let X={a, b, c} and let T ={X, 0, {b}, {a, b}, {b, cn. a) List the closed subsets of (X, T) b) Find Cl({b}) c) Find Cl({a}) d) Find Cl({a, b}) e) Find a proper dense subset ofX.

2.3.3 Show that any finite subset of R is U-closed

2.3.8 Let X=R and let T ={U ex: 1 in U or U =0}. Describe the closed subsets ofX. Find Cl({1,2}). Is {1, 2} a dense subset of (X, T)? (The collection T is an example of the particular point topology.)

2.3.15 Let A and B be subsets of a topological space (X, T). Show that:

X- Cl(A U B) = (X - Cl(A)) n (X - Cl(B)).

Section 2.4: Limit Points} Interior} Boundary} Exterior}

More about Closure

» Let (x, T) be a topological space with a subset A of X if» Limit Points: For all elements x of X, x is said to be a

containing x contain an element of A different from x.

» Interior: The noted Int(A), is the set of all points x in X for which an open set U such that x is in U and U is a subset of A.

» Boundary: The , noted Bd(A), is the set of all points x in X for which containing x intersect both A and X-A.

» Exterior: The noted Ext(A), is the set of all points x in X for which an open set U such that x is in U and U is a subset of X-A.

» Theorem 2.4.4: The set A is closed iff A' is a subset of A » Theorem 2.4.6: The set AUA' is closed » Theorem 2.4.7: CI(A) = AUA' » Theorem 2.4.9: x is in CI(A) iff for any open set U containing x, UnA;t0 » Theorem 2.4.14: Int(A) = U{U subset of A: U is an open set} » Theorem 2.4.15: Int(A) is an open set » Theorem 2.4.16: A is open iff A = Int(A) » Theorem 2.4.18: Ext(A) =Int(X-A) » Theorem 2.4.21: The sets Int(A), Bd(A), and Ext(A) are pairwise disjoint and X =

Int(A)U Bd(A) UExt(A)

» Fact about Closure: CI(A) = Int(A)UBd(A) ./ What is Ext(Ext(A))?

./ How does CI(A) relate to Ext(A)?

./ Let'A = [2,5] U (-1,1) U {9,lD} . What is CI(A)? Bd(A)? Ext(A)? A'? Int(A)?

./ Let X ={a,b,c,d,e,f,g,h}, T ={X, 0, {a,b,c},{c,d}, {C}, {e}, {a,b,c,h}, {a,f,g}, {fH. Let A = {a,d,g,f,h}. What is Int(A)? Bd(A)? Ext(A)? A'? CI(A)?

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4.2 Review Nicholas Otto

1 Carefully define a product space. What is the base for a product space?

2 Let Pi : Xl x Xz x ... X X n -+ Xi be the ith projection function. When is PI a homeomor

phism? pz? When is Pj a homeomorphism for alII :S j :S n? Prove it. Give a simple reason why the projection function is not normally a homeomorphism.

3 Suppose B is a base for Xl x Xz x ... X X n . Construct a base for Xi.

4 Let X = Y = JR, all with the usual topology, Py : X x Y -+ Y be the projection function onto Y and f : X -+ X x Y such that f(x) = (x, sin(x)). Let

Is A with the usual relative topology normal?

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Erin Williams Page 1 of 1

Sec'hon

L Let T be a topology on JR such that VU E T, and for any x E U 3E > a where [x, x + E) C U, let us denote this as JRs. When you take JRsxJRs it is called the Sorgenfrey Half-open square topology. Is JRs normal? Is the Sorgenfrey Half-open square topology normal? If either or both is prove. If they are not give a counterexample.

2. Prove: A space X is regular iff for each x EX, the closed neighborhoods of x form a basis of neighborhoods of x.

3. Give examples of the following topologies on the set X = {a, b, c, d} (not the discrete or indiscrete topologies) .

(a) a topology T such that (X, T) is To space but not T1 space.

(b) a topology S such that (X, S) is T1 space but not T2 space.

(c) a topology R such that (X, R) is T2 space but not T3 space.

(d) a topology Q such that (X, Q) is regular but not normal

(e) a topology P such that (X, P) is normal

4. Prove that T4 property is a topological property.

5. True or False. Prove the true problems and find a counterexample for false.

(a) Any indiscrete topological space is not Hausdorff

(b) Closed subsets of normal spaces are normal

(c) Every T4 space is regular

(d) Every Ti space is a T i - 1 space for each i E {I, 2, 3, 4, 5}

(e) If a topological space does not have any nonempty disjoint open sets, then the space is not normal.