# 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**)