# Equivalence of Definitions of T3 Space

## Contents

## 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$ - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*: Problems: $\S 2: \ 9$