22M:132: Topology Final Exam Dec. 17, 2008 1.) Let R be the set of real numbers and let Z be the set of integers. Let \overline{d}(x, y) = min\{1, |x−y|\}, Identify the following subsets of R.

[3] 1a.) B\overline{d}(0, 1) =

[3] 1b.) \overline{B\overline{d}(0, 1)} =

[3] 1c.) \{x | \overline{d}(x, 0) ≤ 1\} =

[3] 1d.) The set of limit points of Z = Z′ =

[3] 1e.) The closure of Z = \overline{Z} =

Problems 2 and 3 are optional:

[2] 2.) An example of a paracompact space is 3.) Circle T for true and F for false.

[2] 3a.) If X is paracompact, then an arbitrary union of closed sets is closed. T F [2] 3b.) If A is a locally finite collection of closed subsets of X, then ∪_{A∈A}A is closed. T F [80] Prove 4 from the following. Clearly indicate your choices. Note R is the set of real numbers

Your 4 choices:

1.) Let X be a topological space in which one-point sets are closed in X. Show that X is regular if and only if for all x ∈ X, for every open set U in X such that x ∈ U, there is an open set V such that x ∈ V ⊂ \overline{V} ⊂ U.

2i.) Suppose f : X → Y is bijective and continuous, X is compact, and Y is T_{2}. Show that f is a homeomorphism.

ii.) Give an example of a function f : X → Y which is bijective and continuous, but not a homeomorphism where X is a subspace of a manifold and Y is a compact manifold.

3.) A connected, locally pathwise connected space is pathwise connected. (Hint: find a set which is both open and closed).

4.) Recall that if G is a topological group, then m : G × G → G, m(x, y) = xy is continuous. Let G be a topological group and let x, y ∈ G.

i.) Show that for every open neighborhood U of xy, there exists open sets, V and W, such that x ∈ V, y ∈ W and V W ⊂ U.

ii.) If U is an open set containing the identity element e, then there exists an open set V such that e ∈ V and V^{2} = {v_{1}v_{2} | v_{i} ∈ V } ⊂ U.

5.) Every closed subspace of a paracompact space is paracompact.

6.) Let Y^{X} = {f : X → Y }. Let S(x, U) = {f ∈ Y^{X} | f(x) ∈ U}. The topology of pointwise convergence on Y^{X} is the topology generated by the subbasis S = {S(x, U) | x ∈ X, U open in Y }.

i.) S(0, (1, 2) × (1, 2)) ⊂ (R^{2})^{R}. Give an example of a function in S(0, (1, 2) × (1, 2))

ii.) Prove that the sequence f_{n} converges in Y^{X} where Y^{X} has the topology of pointwise convergence if and only if for all x ∈ X, the sequence f_{n}(x) converges to f(x) in Y.

7.) Suppose f : X → R and g : X → R are continuous. i.) Show that {x ∈ X | f(x) = g(x)} is closed in X. ii.) Suppose \{x ∈ X | f(x) = g(x)\} is dense in X (i.e., \overline{\{x ∈ X | f(x) = g(x)\}} = X). Show that f = g.

8.) Let A′ = the set of limit points of A. Determine if the following statements are true. 8i.) (A′)′ ⊂ A′ 8ii.) A′ ⊂ (A′)′.

9.) Every closed subspace of a compact space is compact.

Note you may choose any 4 problems from the above 9 problems (pages 2-3). Problems 1 - 6 are the same on everyone’s final exam. Problems 7 - 9 are additional problems from earlier material.

Your page 1 of the exam will be worth 21 points whether or not you choose to do problems 2 and/or 3. You may do all/part/ or none of problems 2, 3 on page 1 as you choose. Problem 1 on page 1 is required.