Equivalence of Definitions of T3 Space

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Theorem

The following definitions of the concept of $T_3$ space are equivalent:


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

Definition by Open Sets

$T = \left({S, \tau}\right)$ 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 by Closed Neighborhoods

$T = \left({S, \tau}\right)$ 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 by Intersection of Closed Neighborhoods

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

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


Proof

Definition by Open Sets implies Definition by Closed Neighborhoods

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

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



Definition by Closed Neighborhoods implies Definition by Open Sets

$T = \left({S, \tau}\right)$ is a topological space for which:

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




Sources