Separation Axioms

8.1 Overview

Separation axioms formalise how well points and closed sets can be “separated” by open sets.

8.2 $T_0$ (Kolmogorov)

Definition. $X$ is $T_0$ if for any two distinct points $x, y \in X$, at least one has an open neighbourhood not containing the other.

Example 8.1. The Sierpiński space $\{0, 1\}$ with topology $\{\emptyset, \{0\}, \{0, 1\}\}$ is $T_0$ but not $T_1$.

8.3 $T_1$ (Fréchet)

Definition. $X$ is $T_1$ if for any two distinct points $x, y \in X$, there exist open sets $U, V$ with $x \in U$, $y \notin U$ and $y \in V$, $x \notin V$.

Proposition 8.1. $X$ is $T_1$ if and only if every singleton $\{x\}$ is closed.

Example 8.2. Any infinite set with the cofinite topology is $T_1$ but not $T_2$.

8.4 $T_2$ (Hausdorff)

Definition. $X$ is Hausdorff ($T_2$) if for any two distinct points $x, y \in X$, there exist disjoint open sets $U, V$ with $x \in U$ and $y \in V$.

Proposition 8.2. Every metric space is Hausdorff.

Proposition 8.3. In a Hausdorff space, every convergent sequence has a unique limit.

Proposition 8.4. $T_2 \implies T_1 \implies T_0$.

8.5 $T_3$ (Regular) and $T_4$ (Normal)

Definition. $X$ is regular ($T_3$) if it is $T_1$ and for any point $x$ and closed set $F$ with $x \notin F$, there exist disjoint open sets $U, V$ with $x \in U$ and $F \subseteq V$.

Definition. $X$ is normal ($T_4$) if it is $T_1$ and for any two disjoint closed sets $F_1, F_2$, there exist disjoint open sets $U, V$ with $F_1 \subseteq U$ and $F_2 \subseteq V$.

Proposition 8.5. $T_4 \implies T_3 \implies T_2$ (assuming $T_1$).

Proposition 8.6. Every compact Hausdorff space is normal.

8.6 Urysohn”s Lemma

Theorem 8.1 (Urysohn’s Lemma). If $X$ is normal and $F_1, F_2$ are disjoint closed subsets, then there exists a continuous function $f : X \to [0, 1]$ with $f|_{F_1} = 0$ and $f|_{F_2} = 1$.

This is a fundamental tool in topology, used to construct partitions of unity and to prove extension theorems.