---
tag: R3B4V9WW2S
kind: concept
name: Fréchet–Urysohn space
msc: [54D55]
specializes: [WMB2MAEDX4]
sources:
  - url: https://en.wikipedia.org/wiki/Fréchet–Urysohn_space
same_as:
  wikidata: Q96378800
  wikipedia: Fréchet–Urysohn space
---

## Summary

A Fréchet–Urysohn space is a topological space with the following property.
Each point of the closure of a set is the limit of a sequence from that set.
So closures, and not only closed sets, can be computed with sequences.
Every [[first-countable space|0CP5NDD21C]] is Fréchet–Urysohn, and every Fréchet–Urysohn space is [[sequential|WMB2MAEDX4]].
The converse implications fail in general.

## Explanations

### L1 human

In a metric space, a point is in the closure of a set exactly when some sequence from the set runs to it.
A Fréchet–Urysohn space is a space where this familiar fact still holds.
For example, in $\R$ the point $\sqrt 2$ is in the closure of $\Q$, and the decimal approximations $1, 1.4, 1.41, \ldots$ form a sequence of rationals that converges to it.