Definition:T3 Space/Definition 3

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Definition

Let $T = \struct {S, \tau}$ be a topological space.


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

$\forall H \subseteq S: \relcomp S H \in \tau: H = \bigcap \set {N_H: \relcomp S H \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$


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 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.).

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