---
tag: 0R3B7DX55J
kind: definition
name: T0 space
msc: [54D10]
defines: 07SG12TQH3
uses: [0CNQXVBHCX, 01ZQQDCERP]
sources:
  - url: https://en.wikipedia.org/wiki/Kolmogorov_space
    locator: Definition
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is T0 when for all $x \neq y$ in $X$, some [[open set|01ZQQDCERP]] contains exactly one of $x$ and $y$.

## Remarks

Each [[T1 space|05GQJXBP3F]] is T0.
The Sierpiński space $\{0, 1\}$, with open sets $\varnothing$, $\{1\}$ and $\{0, 1\}$, is T0 but not T1.
An [[indiscrete space|0XJ02JBD6S]] with at least two points is not T0.

## Explanations

### L1 human

In a T0 space, no two distinct points have exactly the same open sets around them.
The open sets may still fail to separate the points in both directions.