---
tag: 0D8T984XEE
kind: statement
role: proposition
name: Limits are unique in Hausdorff spaces
msc: [54D10]
uses: [0MXB4CKHFW, 09CJ00H3NW]
sources:
  - url: https://en.wikipedia.org/wiki/Hausdorff_space
    locator: Equivalences
---

## Statement

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]] and let $(x_n)_{n \ge 1}$ be a sequence in $X$.
Say that $(x_n)$ converges to $x \in X$ when for each [[neighbourhood|09CJ00H3NW]] $N$ of $x$ there is $n_0$ with $x_n \in N$ for all $n \ge n_0$.
If $(x_n)$ converges to $x$ and to $y$, then $x = y$.

## Remarks

Proof: if $x \neq y$, take disjoint open sets $U \ni x$ and $V \ni y$. For large $n$, the point $x_n$ is in both $U$ and $V$, which is impossible.
The same holds for nets and filters. In fact, a space is Hausdorff if and only if each net in it has at most one limit.
In an [[indiscrete space|0XJ02JBD6S]] with at least two points, each sequence converges to each point.

## Explanations

### L1 human

A sequence cannot come close to two different points at the same time, if the points have disjoint neighbourhoods.
So in Hausdorff spaces, you can speak of the limit of a sequence.