Completely Regular and Tychonoff Spaces

Completely regular spaces are those where points and closed sets can be separated by continuous functions. Combined with the $T_1$ axiom, they yield Tychonoff spaces ( $T_{3\frac{1}{2}}$ ), which are precisely the spaces that embed into products of the unit interval and are the natural domain for the Stone--Cech compactification.

Definition

Definition6.6Completely Regular Space

A topological space $X$ is completely regular if for every point $x \in X$ and every closed set $C$ with $x \notin C$ , there exists a continuous function $f: X \to [0, 1]$ such that $f(x) = 0$ and $f|_C \equiv 1$ .

Definition6.7Tychonoff Space ($T_{3\frac{1}{2}}$)

A space is Tychonoff (or $T_{3\frac{1}{2}}$ or completely regular Hausdorff) if it is completely regular and $T_1$ .

RemarkPosition in the Hierarchy

$T_4 \implies T_{3\frac{1}{2}} \implies T_3 \implies T_2 \implies T_1 \implies T_0$

The implication $T_4 \implies T_{3\frac{1}{2}}$ follows from the Urysohn lemma: in a normal $T_1$ space, disjoint closed sets (and in particular a point and a closed set) can be separated by a continuous function to $[0,1]$ .

The implication $T_{3\frac{1}{2}} \implies T_3$ holds because if $f: X \to [0,1]$ separates $x$ from $C$ , then $f^{-1}([0, 1/2))$ and $f^{-1}((1/2, 1])$ are disjoint open sets separating $x$ from $C$ .

Characterization via Embeddings

Theorem6.6Embedding Theorem for Tychonoff Spaces

A topological space $X$ is Tychonoff if and only if $X$ embeds into a product of copies of $[0, 1]$ . Equivalently, $X$ is Tychonoff if and only if $X$ embeds into some compact Hausdorff space.

Proof

( $\Rightarrow$ ): Let $\mathcal{F} = C(X, [0,1])$ be the set of continuous functions $X \to [0,1]$ . Define $e: X \to [0,1]^{\mathcal{F}}$ by $e(x) = (f(x))_{f \in \mathcal{F}}$ . We claim $e$ is an embedding.

Injective: If $x \neq y$ , since $X$ is $T_1$ , $\{y\}$ is closed. Since $X$ is completely regular, there exists $f \in \mathcal{F}$ with $f(x) = 0$ and $f(y) = 1$ . So $e(x) \neq e(y)$ .

Continuous: The composition $\pi_f \circ e = f$ is continuous for each $f$ , so $e$ is continuous by the universal property of products.

Open onto image: For open $U \subseteq X$ and $x \in U$ , there exists $f \in \mathcal{F}$ with $f(x) = 0$ and $f|_{X \setminus U} \equiv 1$ . Then $e(U) \supseteq e(X) \cap \pi_f^{-1}([0, 1/2))$ , showing $e(U)$ is open in $e(X)$ .

( $\Leftarrow$ ): Subspaces of compact Hausdorff spaces are Tychonoff. (Compact Hausdorff implies normal, and subspaces of normal $T_1$ spaces are completely regular and $T_1$ .)

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Examples

ExampleTychonoff Spaces

All metric spaces are Tychonoff. The function $f(y) = \min(d(y, C)/d(x, C), 1)$ separates $x$ from a closed set $C$ .
All locally compact Hausdorff spaces are Tychonoff (they are regular, and regularity combined with local compactness gives complete regularity).
Subspaces and products of Tychonoff spaces are Tychonoff.
Topological groups that are $T_0$ are Tychonoff (a result due to Pontryagin).

ExampleRegular but Not Completely Regular

There exist $T_3$ spaces that are not $T_{3\frac{1}{2}}$ (the Tychonoff corkscrew and other pathological examples), but they are quite exotic. In practice, all commonly encountered regular spaces are completely regular.

The Stone--Cech Compactification Preview

Definition6.8Stone--Cech Compactification

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

RemarkWhy Tychonoff is the Right Condition

The Stone--Cech compactification $\beta X$ exists for a space $X$ if and only if $X$ is Tychonoff. This is because:

Tychonoff spaces embed into compact Hausdorff spaces (the embedding theorem above).
The closure of $X$ in $[0,1]^{\mathcal{F}}$ gives a compactification.
The universal property makes this the "largest" compactification.

The class of Tychonoff spaces is therefore the natural domain for compactification theory. See Chapter 8 for the full treatment.

Functional Separation

Theorem6.7Functional Separation in Tychonoff Spaces

In a Tychonoff space $X$ , the topology is completely determined by the ring $C(X, \mathbb{R})$ of continuous real-valued functions. More precisely:

$U \subseteq X$ is open if and only if for every $x \in U$ , there exists $f \in C(X, \mathbb{R})$ with $f(x) > 0$ and $f|_{X \setminus U} \equiv 0$ .
The zero sets $Z(f) = f^{-1}(0)$ for $f \in C(X, \mathbb{R})$ form a basis for the closed sets.

ExampleZero Sets

In $\mathbb{R}$ , any closed set is a zero set of some continuous function: if $C$ is closed, define $f(x) = d(x, C) = \inf_{c \in C} |x - c|$ , which is continuous and $f^{-1}(0) = C$ .

In a general Tychonoff space, closed sets need not be zero sets, but they are intersections of zero sets.