---
tag: 04J4BYB0XE
kind: concept
name: Separation axioms
aliases: [tychonoff-separation-axioms]
msc: [54D10]
sources:
  - url: https://en.wikipedia.org/wiki/Separation_axiom
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Separation_axiom
    locator: Main definitions
same_as:
  wikidata: Q1140985
  wikipedia: Separation axiom
---

## Summary

The separation axioms are conditions on a topological space that say how well open sets can separate points and closed sets.
The main ones, from weak to strong, are [[T0|0R3B7DX55J]], [[T1|05GQJXBP3F]], [[Hausdorff|0MXB4CKHFW]], [[regular|0JF9KW2TXB]] and [[normal|0AJ646N339]].

## Remarks

Each of these axioms, together with T1, implies the axioms before it in this list.
Each metric space satisfies all of them, by [[metric spaces are Hausdorff|0P9DCDES73]] and [[metrizable spaces are normal|0AD0BZ49DW]].
Authors do not agree on all names. For example, some authors include T1 in the definition of a regular or normal space, and some do not.

## Explanations

### L1 human

The separation axioms measure how many open sets a space has.
With more open sets, you can put distinct points, or points and closed sets, into disjoint open sets.