# Frechet–Urysohn spaces

## Idea

A Frechet–Urysohn (or Frechet–Uryson) space is a topological space in which the closure of a subspace may be described using only sequences.

## Definition

Recall that, given any subset $A$ of any topological space, a point $x$ belongs to the closure of $A$ if and only if $x$ is a limit point of at least one net whose elements belong to $A$.

A topological space is Frechet–Uryson (or Frechet–Urysohn) if a point $x$ of the closure of any given subset $A$ of $X$ is a limit point of at least one sequence whose elements belong to $A$.

## Examples

Every first-countable space is a Frechet–Uryson space.

## Properties

Every Frechet–Uryson space is a sequential space.

Revised on December 13, 2009 03:43:45 by Toby Bartels (173.60.119.197)