## Warm-up Problems

• If X is $T _ { 2 }$ and $A \subseteq X$ is compact, then A is closed.

• If X is compact and $A \subseteq X$ is closed, then A is compact.

• (June 2011, A1) If $f : X \to Y$ is continuous and $A \subseteq X$ is compact, then $f ( A )$ is compact.

1. (Jan 2002, A1) Let X be a compact space and let

$$
A _ { 1 } \supseteq A _ { 2 } \supseteq \cdots \supseteq A _ { n } \supseteq \cdots
$$

be a descending chain of nonempty closed subsets of X. Show that their intersection $\cap _ { n = 1 } ^ { \infty } A _ { n }$ is nonempty.

2. (June 2005, A2) Let X be the set of integers, let ${ \mathcal { C } } _ { 1 } : = \{ A \subseteq X | X - A$ is finite}, and let $\mathcal { C } _ { 2 } : = \{ A \subset X | 0 \notin A \}$ . Show that the union $\mathcal { T } : = \mathcal { C } _ { 1 } \cup \mathcal { C } _ { 2 }$ is a topology on X, and show that this topological space is compact.

3. (Jan 2006, A4) Let $\mathcal { T } , \mathcal { T } ^ { \prime }$ be two topologies on X. Show that if $( X , \tau )$ is compact and Hausdorff, $\tau \subseteq \tau ^ { \prime }$ , and $\mathcal { T } \neq \mathcal { T } ^ { \prime }$ , then $( X , { \mathcal { T } } ^ { \prime } )$ is Hausdorff but not compact.

4. (June 2008, A4) Let $Y = ( 0 , 1 ) \times ( 0 , 1 )$ and let X be the set $X = Y \cup \{ \star \}$ . Let $\mathcal { T } _ { 1 } : =$ $\{ U | U$ is a Euclidean open subset of Y }, let ${ \mathcal { T } } _ { 2 } : = \{ X - C | C$ is a compact subset of the Euclidean space Y }, and let $\mathcal { T } : = \mathcal { T } _ { 1 } \cup \mathcal { T } _ { 2 }$ . Then (X, T ) is a topological space called the one-point compactification of $( 0 , 1 ) \times ( 0 , 1 )$ (you don’t need to prove this). Show that the space $( X , \tau )$ is compact.

5. (June 2009, A4) Let $( X , \tau )$ and $( X , { \mathcal { T } } ^ { \prime } )$ be topological spaces with $\tau \subseteq \tau ^ { \prime }$ . [(a) omitted] (b) If $( X , { \mathcal { T } } ^ { \prime } )$ is compact, must $( X , \tau )$ also be compact?

6. (June 2010, A2) Let X be a Hausdorff space, $p \in X$ , and $A \subseteq X$ a compact subset of X disjoint from p. Show that there exist disjoint open sets $U , V \subseteq X$ with $p \in U$ and $A \subseteq V$

7. (June 2011, A4) Suppose A, B are disjoint, compact subspaces of the Hausdorff topological space X. Prove that there are open subsets U, V of X such that $A \subseteq U , B \subseteq V$ and $U \cap V = \emptyset$

8. (June 2004, A1; June 2007, A4) A space $( X , \tau )$ is called locally compact if, for every $x \in X$ and $x \in U \in \mathcal { T }$ , there is a compact set $C \subseteq U$ containing an open neighborhood of x. Show that the Cartesian product of two locally compact spaces is locally compact.

9. Suppose that X is compact and Y is Hausdorff. Prove that every one-to-one, onto, continuous map $f : X \to Y$ is a homeomorphism.