---
tag: 05GQJXBP3F
kind: definition
name: T1 space
msc: [54D10]
defines: 0NW6CH38YR
uses: [0CNQXVBHCX, 01ZQQDCERP, 0G5AV9Q4RR]
sources:
  - url: https://en.wikipedia.org/wiki/T1_space
    locator: Definitions
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is T1 when for all points $x \neq y$ of $X$ there is an [[open set|01ZQQDCERP]] $U$ with $x \in U$ and $y \notin U$.

## Remarks

A space is T1 if and only if each set $\{x\}$ with $x \in X$ is [[closed|0G5AV9Q4RR]].
Each [[Hausdorff space|0MXB4CKHFW]] is T1, and each T1 space is [[T0|0R3B7DX55J]].
An infinite set with the [[cofinite topology|0B906ZBTMZ]] is T1 but not Hausdorff. In fact, the cofinite topology is the coarsest T1 topology on a set.

## Explanations

### L1 human

In a T1 space, for two distinct points, each point has an open set around it that does not contain the other point.
This is the same as saying that single points are closed.