characterization of $T2$ spaces
Proposition 1.
[1, 2] Suppose $X$ is a topological space^{}. Then $X$ is a ${T}_{\mathrm{2}}$ space (http://planetmath.org/T2Space) if and only if for all $x\mathrm{\in}X$, we have
$\mathrm{\{}x\}$ | $=$ | $\bigcap \{A\mid A\subseteq X\text{\mathit{c}\mathit{l}\mathit{o}\mathit{s}\mathit{e}\mathit{d}},\exists \mathit{\text{open set}}U\mathit{\text{such that}}x\in U\subseteq A\}}.$ | (1) |
Proof.
By manipulating the definition using de Morgan’s laws, the claim can be rewritten as
$${\{x\}}^{\mathrm{\complement}}=\bigcup \{V\mid V\subseteq X\text{open},\exists \text{open set}U\text{such that}x\in U\subseteq {V}^{\mathrm{\complement}}\}.$$ |
Suppose $y\in {\{x\}}^{\mathrm{\complement}}$. As $X$ is a ${T}_{2}$ space, there are open sets $U,V$ such that $x\in U,y\in V$, and $U\cap V=\mathrm{\varnothing}$. Thus, the inclusion from left to right holds. On the other hand, suppose $y\in V$ for some open $V$ such that $\{x\}\subseteq {V}^{\mathrm{\complement}}$. Then
$$y\in V\subseteq {\{x\}}^{\mathrm{\complement}}$$ |
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
$\mathrm{\{}x\}$ | $=$ | $\bigcap \{A\mid A\subseteq X\text{is a closed neighborhood of}x\}}.$ |
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 |
---|---|
Canonical name | CharacterizationOfT2Spaces |
Date of creation | 2013-03-22 14:41:47 |
Last modified on | 2013-03-22 14:41:47 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 7 |
Author | matte (1858) |
Entry type | Theorem^{} |
Classification | msc 54D10 |
Related topic | LocallyCompactHausdorffSpace |