Definition:T3 Space

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

Definition 1

$T = \left({S, \tau}\right)$ 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 $F \subseteq U$, $y \in V$.

Definition 2

$T = \left({S, \tau}\right)$ is $T_3$ if and only if each open set contains a closed neighborhood around each of its points:

$\forall U \in \tau: \forall x \in U: \exists N_x: \complement_S \left({N_x}\right) \in \tau: \exists V \in \tau: x \in V \subseteq N_x \subseteq U$

Definition 3

$T = \left({S, \tau}\right)$ is $T_3$ if and only if each of its closed sets is the intersection of its closed neighborhoods:

$\forall H \subseteq S: \complement_S \left({H}\right) \in \tau: H = \bigcap \left\{{N_H: \complement_S \left({N_H}\right) \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}\right\}$

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.