---
tag: C5N2J2K6Z3
kind: concept
name: Tychonoff space
msc: [54D15]
specializes: [03HMJASQBQ]
sources:
  - url: https://en.wikipedia.org/wiki/Tychonoff_space
same_as:
  wikidata: Q1191433
  wikipedia: Tychonoff space
---

## Summary

A Tychonoff space is a [[T1 space|0NW6CH38YR]] with many continuous real functions.
Every point can be separated by such a function from every closed set that does not contain it.
Such spaces are exactly the subspaces of products of copies of $[0, 1]$, up to homeomorphism.
They are also exactly the spaces that embed in some compact Hausdorff space.
Metric spaces and compact Hausdorff spaces are Tychonoff.

## Explanations

### L1 human

In a Tychonoff space, there are enough continuous real functions to describe the topology.
Given a point and a closed set away from it, there is a function that is $0$ at the point and $1$ on the set.
You can think of such a function as a continuous hill that rises from the point to the set.
In the plane, for a point $p$ and a closed set $F$ with $p \notin F$, the function $\min(1, d(x, p)/d(p, F))$ does the job.