Definition:Tychonoff Space

Definition

Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a Tychonoff Space or completely regular space if and only if:

$\struct {S, \tau}$ is a $T_{3 \frac 1 2}$ space

$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.

That is:

For any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\set y$.

$\forall x, y \in S$, either:

$\exists U \in \tau: x \in U, y \notin U$

$\exists U \in \tau: y \in U, x \notin U$

Variants of Name

From about 1970, treatments of this subject started to refer to this as a $T_{3 \frac 1 2}$ space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a $T_{3 \frac 1 2}$ space as a completely regular space.

However, the names are to a fair extent arbitrary and a matter of taste, as there appears to be no completely satisfactory system for naming all these various Tychonoff separation axioms.

The system as used here broadly follows 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.).

The system used on the Separation axiom page at Wikipedia differs from this.

Also defined as

Some sources define a Tychonoff space as a topological space which is both completely regular and Hausdorff.

Also known as

Some sources give this as $T_{\frac 7 2}$ space, which of course evaluates to the same as a $T_{3 \frac 1 2}$ space.

Also see

• Results about Tychonoff spaces can be found here.

Source of Name

This entry was named for Andrey Nikolayevich Tychonoff.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Regular Spaces
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): T-axioms or Tychonoff conditions: 3b. ($T_{\frac 7 2}$ space or Tychonoff Space)