Warm-up Problems:

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

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

• (June $^ { \mathrm { ~ \small ~ } } 1 1 \neq \mathrm { ~ A 1 } )$ If $f : X \to Y$ is continuous and $A \subseteq X$ is compact, then $f \left( A \right)$ is compact.

(1) (Jan ${ \bf \Phi } ^ { \mathrm { 3 } } 0 2 \# { \bf A } 1 )$ 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 non-empty closed subsets of X. Show that their intersection $\cap _ { n = 1 } ^ { \infty } A _ { n }$ is non-empty.

(2) (June ’04 # A1 [and June $^ { \ ' } 0 7 \ \# \ \mathrm { A } 4 ] )$ A topological space X is locally compact if every point in X has an open neighborhood whose closure is compact.
Show that the Cartesian product of two locally compact spaces, with the product topology, is also locally compact.

(3) (June ${ \bf \ ' } 0 5 \not = \mathrm { A 2 } )$ Let X be the set of integers, let ${ \mathcal { C } } _ { 1 } : = \{ A \subseteq X | X \setminus A$ is finite }, and let $\mathcal { C } _ { 2 } : = \{ A \subseteq X | 0 \not \in A \}$ . 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.

(4) (Jan ${ } ^ { \ ' } 0 6 \not = \mathrm { A } 4 )$ Let $\tau , \tau ^ { \prime }$ be two topologies on X. Show that if $( X , \tau )$ is compact and Hausdorff, $\mathcal { T } \subseteq \mathcal { T } ^ { \prime }$ , and $\tau \neq \tau ^ { \prime }$ , then $( X , T ^ { \prime } )$ is Hausdorff but not compact.

(5) (June ${ \bf \ ' 0 5 \# A 4 ) }$ A topological space X is called metacompact if for every open cover C of X, there is a subcover $C ^ { \prime }$ satisfying the property that for every point $p \in X$ , there are only finintely many open sets in $C ^ { \prime }$ containing p. w

(a) Show that metacompactness is a homeomorphism invariant.

(b) Let X be the integers with the topology $\tau = \{ U \subset X : 0 \in U \} \cup \{ \emptyset \}$ . Show that this space is not metacompact.

(6) (June ’11 $\# \ : \mathrm { A 4 } )$ 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 $U \cap V = \emptyset$

(7) 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.

(8) (Purdue ’11) Let X be a set with two elements $\{ a , b \}$ . Give X the indiscrete topology.
Give $X \times \mathbb { R }$ the product topology.
Let $A \subset X \times \mathbb { R }$ be $( \{ a \} \times [ 0 , 1 ] ) \cup ( \{ b \} \times ( 0 , 1 ) )$ . Prove that A is compact.

(9) (Purdue ’11) Let X be a compact space and let $\{ C _ { \alpha } \} _ { \alpha \in A }$ be a collection of closed sets in X. Let $\textstyle C = \cap _ { \alpha \in A } C _ { \alpha }$ and let U be an open set containing C. Prove there is a finite set $\alpha _ { 1 } , . . . \alpha _ { n }$ in A with $C _ { \alpha _ { 1 } } \cap \cdot \cdot \cdot \cap C _ { \alpha _ { n } } \subset U$