## Warm-up Problems:

$\operatorname { I f } \ ( X , d )$ is a metric space, then it is Hausdorff [see number 8].

$$
T _ { 4 } \implies T _ { 3 } \implies T _ { 2 } \implies T _ { 1 } \implies T _ { 0 }
$$

• Suppose $( X , \tau )$ is a topological space and let $( X , \tau ^ { \prime } )$ be same set with the finite complement topology.
Then X is $T _ { 1 }$ if and only if id $: ( X , \tau ) \ \to \ ( X , \tau ^ { \prime } )$ is continuous.

(1) (Jan $^ { \prime } 0 8 \# \mathrm { B 9 } )$ Let $X = \mathbb { R } ^ { 2 }$ and define an equivalence relation on X by $( x _ { 1 } , x _ { 2 } ) \sim ( y _ { 1 } , y _ { 2 } )$ iff they are qual or $x _ { 1 } = y _ { 1 } = 0$ . Set $Y = X / \sim .$ Show that Y is Hausdorff but does not have a countable basis for its topology.

(2) (June $0 6 \neq \mathrm { A 5 } )$

(a) Recall that a topological space is called separable if it contains a countable dense subset.
Prove that a countable product of separable spaces (with the product topology is separable).

(b) Prove that the result in the first part is false if the product is given the box topology.

(3) (June $\mathrm { \Delta 0 4 \# \ A 3 ) }$ Show that for a topological space X, if every $x \in X$ has a neighborhood whose closure is a regular space, then X is regular.

(4) (Jan $^ { \prime } 0 4 \ \# \ \mathrm { A 2 } )$ Show that if the metric space $( X , d )$ is separable, then the metric topology on X is second countable.

(5) (Jan $\mathbf { 0 } 4 \# \mathrm { A } 4 )$ Two subsets $A , B \subseteq X$ of the space $( X , \tau )$ are called separated if there are $U , V \in \tau$ with $A \subseteq U \subseteq X \setminus B$ and $B \subseteq V \subseteq X \setminus A$ We say that X is completely normal if X is $T _ { 1 }$ T and if for every pair of separated subsets A, B there are $U , V \in \tau$ so that $A \subseteq U , B \subseteq V$ and $U \cap V = \emptyset$ . Show that a space $( X , \tau )$ is completely normal ⇔ every subset of X is normal.

(6) (Jan $^ { \ ' } 1 2 \# \mathrm { A } 2 )$ Let X and Y be two topological spaces and let $X \times Y$ be endowed with the product topology.
Prove that if X and Y each have a countable dense subset, then so does $X \times Y$

(7) (Jan $^ { 1 2 } \# \mathrm { A } 4 )$

(a) State the definition of a what it means for a topological space to be regular.

(b) Prove that a subspace of a regular space is also regular.

(c) Prove that a product of two regular spaces (equipped with the product topology) is also regular.

(8) (June ’11 $\# \mathrm { A 3 } )$ Prove that every metrizable space is normal Hausdorff $\left( \mathrm { a k a } T _ { 4 } \right)$

(9) (June ${ \bf \Phi } ^ { 0 9 } \# \mathrm { { A 4 } ) }$ Let $( X , \tau )$ and $( X , \tau ^ { \prime } )$ be topological spaces with $\tau \subseteq \tau ^ { \prime }$

(a) If $( X , \tau ^ { \prime } )$ is normal, must $( X , \tau )$ also be normal?

(b) If $( X , \tau ^ { \prime } )$ is compact, must $( X , \tau )$ also be compact?