Compact Hausdorff Spaces

Compact Hausdorff spaces form one of the most well-behaved classes of topological spaces. They are automatically normal, their topology is "rigid" (neither coarsened nor refined while remaining compact Hausdorff), and many strong theorems hold for them.

Separation in Compact Hausdorff Spaces

Theorem5.10Compact Hausdorff Spaces are Normal

Every compact Hausdorff space is normal. More precisely:

Compact Hausdorff $\Rightarrow$ regular ( $T_3$ ).
Compact Hausdorff $\Rightarrow$ normal ( $T_4$ ).

Proof

(1): Let $X$ be compact Hausdorff, $x \in X$ , and $C$ closed with $x \notin C$ . Since $C$ is closed in a compact space, $C$ is compact. For each $c \in C$ , the Hausdorff property gives disjoint open sets $U_c \ni x$ and $V_c \ni c$ . The sets $\{V_c\}_{c \in C}$ cover $C$ ; extract a finite subcover $V_{c_1}, \ldots, V_{c_n}$ . Set: $U = U_{c_1} \cap \cdots \cap U_{c_n}, \qquad V = V_{c_1} \cup \cdots \cup V_{c_n}.$ Then $U \ni x$ , $V \supseteq C$ , both are open, and $U \cap V = \emptyset$ .

(2): Let $C, D$ be disjoint closed subsets. Both are compact. For each $c \in C$ , part (1) gives disjoint open sets $U_c \supseteq D$ and $V_c \ni c$ . The sets $\{V_c\}_{c \in C}$ cover $C$ ; extract a finite subcover $V_{c_1}, \ldots, V_{c_m}$ . Set: $U = U_{c_1} \cap \cdots \cap U_{c_m}, \qquad V = V_{c_1} \cup \cdots \cup V_{c_m}.$ Then $U \supseteq D$ , $V \supseteq C$ , both are open, and $U \cap V = \emptyset$ .

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Uniqueness of the Compact Hausdorff Topology

Theorem5.11Rigidity of Compact Hausdorff Topologies

Let $\tau$ be a compact Hausdorff topology on a set $X$ .

If $\tau' \subsetneq \tau$ (strictly coarser), then $(X, \tau')$ is not Hausdorff.
If $\tau \subsetneq \tau'$ (strictly finer), then $(X, \tau')$ is not compact.

In other words, a compact Hausdorff topology is minimal Hausdorff and maximal compact.

Proof

(1): Suppose $\tau'$ is Hausdorff. The identity map $\operatorname{id}: (X, \tau) \to (X, \tau')$ is continuous (since $\tau' \subseteq \tau$ ) and bijective. Since $(X, \tau)$ is compact and $(X, \tau')$ is Hausdorff, $\operatorname{id}$ is a homeomorphism (Theorem 2.11). Thus $\tau = \tau'$ .

(2): Suppose $(X, \tau')$ is compact. The identity $\operatorname{id}: (X, \tau') \to (X, \tau)$ is continuous (since $\tau \subseteq \tau'$ ), bijective, from compact to Hausdorff. So it is a homeomorphism, giving $\tau' = \tau$ .

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Continuous Functions on Compact Hausdorff Spaces

Theorem5.12$C(X)$ Separates Points

If $X$ is a compact Hausdorff space, then the ring of continuous functions $C(X, \mathbb{R})$ separates points: for distinct $x, y \in X$ , there exists $f \in C(X, \mathbb{R})$ with $f(x) \neq f(y)$ .

Proof

By normality and the Urysohn lemma (Theorem 6.1), for disjoint closed sets $\{x\}$ and $\{y\}$ , there exists a continuous function $f: X \to [0, 1]$ with $f(x) = 0$ and $f(y) = 1$ .

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Theorem5.13Closed Map Theorem

Let $X$ be compact and $Y$ Hausdorff. Every continuous map $f: X \to Y$ is a closed map.

Proof

Let $C \subseteq X$ be closed. Then $C$ is compact (closed subset of compact). So $f(C)$ is compact (continuous image of compact). Since $Y$ is Hausdorff, $f(C)$ is closed.

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Perfect Maps

Definition5.10Perfect Map

A continuous map $f: X \to Y$ is perfect if it is a closed surjection with compact fibers (i.e., $f^{-1}(y)$ is compact for every $y \in Y$ ).

Theorem5.14Perfect Maps Preserve Compactness

Let $f: X \to Y$ be a perfect map. Then $X$ is compact if and only if $Y$ is compact.

ExamplePerfect Maps

Any continuous map from a compact space to a Hausdorff space is perfect (it is closed, surjective onto its image, and fibers are closed subsets of a compact space hence compact).
The quotient map $q: S^n \to \mathbb{R}P^n$ (identifying antipodal points) is a perfect map with each fiber consisting of exactly two points.

RemarkStone--Cech and Compact Hausdorff Spaces

Compact Hausdorff spaces form a reflective subcategory of the category of Tychonoff spaces. The reflector is the Stone--Cech compactification $\beta$ . For a Tychonoff space $X$ , $\beta X$ is the "largest" compact Hausdorff space containing $X$ as a dense subspace. This functorial construction plays a fundamental role in functional analysis and set-theoretic topology.