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# Properties of Hausdorff Spaces I

TOPO-KX7WGQ

Let $(X,\tau)$ be a topological space.

Which of the following statements gives the definition of $(X,\tau)$ being a Hausdorff space?

A

For all $x,y\in X$ with $x\neq y$, there exists $U\in\tau$ with $x\in U$ and $y\notin U$.

B

For all $x,y\in X$ with $x\neq y$, there exists $U,V\in\tau$ with $x\in U$, $y\in V$, $y\notin U$, and $x\notin V$.

C

For all $x,y\in X$ with $x\neq y$, there exists $U,V\in\tau$ with $x\in U$, $y\in V$, and $U\cap V=\emptyset$.

D

For all $x,y\in X$ with $x\neq y$, there exists closed sets $A$ and $B$ with $x\in A$, $y\in B$, and $A\cap B=\emptyset$.

E

For all $x\in X$ and every nonempty closed set $A$ with $x\not\in A$, there exist $U,V\in\tau$ with $x\in U$, $A\subseteq V$, and $U\cap V=\emptyset$.