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Topology

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Properties of Hausdorff Spaces II

TOPO-KQEXKY

Let $(X,\tau)$ be a Hausdorff space. Which of the following statements MUST be true?

Select ALL that apply.

A

$\tau$ is the metric topology induced by some metric.

B

For every distinct $x,y\in X$, there exist disjoint open sets $U$ and $V$ with $x\in U$ and $y\in V$.

C

For every $x\in X$, $\{x\}$ is a closed set.

D

For every distinct closed sets $A$ and $B$, there exist distjoint open sets $U$ and $V$ with $A\subseteq U$ and $B\subseteq V$.

E

$\tau$ is the discrete topology.