Definition:T3 Space/Definition 1

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.


$T = \left({S, \tau}\right)$ is a $T_3$ space if and only if:

$\forall F \subseteq S: \complement_S \left({F}\right) \in \tau, y \in \complement_S \left({F}\right): \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \varnothing$

That is, for any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$ there exist disjoint open sets $U, V \in \tau$ such that $F \subseteq U$, $y \in V$.


That is:

$\left({S, \tau}\right)$ is $T_3$ when any closed set $F \subseteq S$ and any point not in $F$ are separated by neighborhoods.


Variants of Name

From about 1970, treatments of this subject started to refer to this as a regular space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a regular space as a $T_3$ space.

However, the names are to a fair extent arbitrary and a matter of taste, as there appears to be no completely satisfactory system for naming all these various Tychonoff separation axioms.


The system as used here broadly follows 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology.

The system used on the Separation axiom page at Wikipedia differs from this.


Also see

  • Results about $T_3$ spaces can be found here.


Sources