Separation Axioms

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

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 $T_1$.

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 $T_2$.

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. $T_2 \implies T_1 \implies T_0$.

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.

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.

Definition. A space $X$ is completely regular (Tychonoff, $T_{3.5}$) if it is $T_1$ and for every point $x$ and closed set $F$ with $x \notin F$, there exists a continuous function $f : X \to [0, 1]$ with $f(x) = 0$ and $f(F) = 1$.

Proposition 8.7. Every normal space is completely regular. Every completely regular space is regular: $T_4 \implies T_{3.5} \implies T_3$.

Example 8.3. The Sorgenfrey line (lower limit topology on $\mathbb{R}$) is Tychonoff but not normal.

Theorem 8.2 (Tietze Extension Theorem). If $X$ is normal, $F \subseteq X$ is closed, and $f : F \to [a, b]$ is continuous, then $f$ extends to a continuous function $\tilde{f} : X \to [a, b]$ with $\tilde{f}|_F = f$.

This follows from Urysohn’s lemma by constructing a sequence of approximations whose sum converges uniformly to the extension.

Theorem 8.3 (Urysohn Metrization Theorem). Every second-countable regular $T_1$ space is metrizable (its topology is induced by some metric).

Proof sketch. Embed $X$ into the Hilbert cube $[0, 1]^\omega$ using a countable collection of Urysohn functions constructed from a countable basis. The Hilbert cube is metrizable, so the subspace topology on $X$ is metrizable. $\blacksquare$

Corollary 8.8. A space is a separable metric space if and only if it is second-countable and regular $T_1$.

The hierarchy of separation axioms:

$T_4 \implies T_{3.5} \implies T_3 \implies T_2 \implies T_1 \implies T_0$

Each implication is strict: there exist spaces satisfying each level but not the next.

Problem 1. Show that the Zariski topology on $\mathbb{R}^n$ is $T_1$ but not $T_2$.

Problem 2. Prove that a closed subspace of a normal space is normal.

Solution. Let $X$ be normal and $F \subseteq X$ closed. Let $A, B$ be disjoint closed sets in $F$. Then $A, B$ are also closed in $X$ (since $F$ is closed). By normality of $X$, there exist disjoint open $U, V \subseteq X$ with $A \subseteq U$, $B \subseteq V$. Then $U \cap F$ and $V \cap F$ are disjoint open in $F$ separating $A$ and $B$. $\blacksquare$

Problem 3. Show that every metric space is normal.

Problem 4. Prove that a product of two $T_2$ spaces is $T_2$. Is the same true for $T_3$ or $T_4$?

Problem 5. Show that the Sorgenfrey plane $\mathbb{R}_\ell \times \mathbb{R}_\ell$ is not normal, even though the Sorgenfrey line $\mathbb{R}_\ell$ is normal. (Hint: the antidiagonal $\{(x, -x) : x \in \mathbb{R}\}$ is closed and discrete.)

Definition. A space $X$ is completely normal ($T_5$) if every subspace of $X$ is normal. Equivalently, for any two separated sets $A, B$ (i.e., $\bar{A} \cap B = A \cap \bar{B} = \emptyset$), there exist disjoint open sets separating them.

Definition. A space $X$ is perfectly normal ($T_6$) if $X$ is normal and every closed set is a $G_\delta$ set (a countable intersection of open sets).

Proposition 8.9. Every metric space is $T_6$ (perfectly normal).

Proposition 8.10. $T_6 \implies T_5 \implies T_4$ (assuming $T_1$).

Theorem 8.4 (Tychonoff Product Theorem). The product of any collection of compact topological spaces is compact (in the product topology).

This is one of the most important theorems in general topology, equivalent to the axiom of choice. The proof uses the finite intersection property and Zorn’s lemma.

For a Tychonoff space $X$, the Stone-Cech compactification $\beta X$ is the unique compact Hausdorff space containing $X$ as a dense subspace such that every continuous map from $X$ to a compact Hausdorff space extends continuously to $\beta X$.

Example. $\beta \mathbb{N}$ is the set of ultrafilters on $\mathbb{N}$ with the topology generated by $\{A^* : A \subseteq \mathbb{N}\}$ where $A^* = \{U \in \beta\mathbb{N} : A \in U\}$.

Problem 6. Prove that every compact subset of a Hausdorff space is closed.

Problem 7. Show that $\mathbb{Q}$ with the subspace topology from $\mathbb{R}$ is not a $T_4$ space.

Problem 8. Prove that a space is Hausdorff if and only if the diagonal $\Delta = \{(x, x) : x \in X\}$ is closed in $X \times X$.