nLab
Frechet-Uryson space

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.