# characterization of $T2$ spaces

###### Proposition 1.

[1, 2] Suppose $X$ is a topological space. Then $X$ is a $T_{2}$ space (http://planetmath.org/T2Space) if and only if for all $x\in X$, we have

 $\displaystyle\{x\}$ $\displaystyle=$ $\displaystyle\bigcap\{A\mid A\subseteq X\ \mbox{closed},\mbox{\exists open % set}\ U\ \mbox{such that}\ x\in U\subseteq A\}.$ (1)
###### Proof.

By manipulating the definition using de Morgan’s laws, the claim can be rewritten as

 $\{x\}^{\complement}=\bigcup\{V\mid V\subseteq X\ \mbox{open},\mbox{\exists % open set}\ U\ \mbox{such that}\ x\in U\subseteq V^{\complement}\}.$

Suppose $y\in\{x\}^{\complement}$. As $X$ is a $T_{2}$ space, there are open sets $U,V$ such that $x\in U,y\in V$, and $U\cap V=\emptyset$. Thus, the inclusion from left to right holds. On the other hand, suppose $y\in V$ for some open $V$ such that $\{x\}\subseteq V^{\complement}$. Then

 $y\in V\subseteq\{x\}^{\complement}$

and the claim follows. ∎

## Notes

If we adopt the notation that a neighborhood of $x$ is any set containing an open set containing $x$, then the equation 1 can be written as

 $\displaystyle\{x\}$ $\displaystyle=$ $\displaystyle\bigcap\{A\mid A\subseteq X\ \mbox{is a closed neighborhood of x% }\}.$

## References

• 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
• 2 N. Bourbaki, General Topology, Part 1, Addison-Wesley Publishing Company, 1966.
Title characterization of $T2$ spaces CharacterizationOfT2Spaces 2013-03-22 14:41:47 2013-03-22 14:41:47 matte (1858) matte (1858) 7 matte (1858) Theorem msc 54D10 LocallyCompactHausdorffSpace