# Definition:Fréchet Space (Topology)

This page is about the Fréchet topological space, characterised by the fact that all points are closed. For other uses, see Definition:Fréchet Space.

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

### Definition 1

$\left({S, \tau}\right)$ is a Fréchet space or $T_1$ space if and only if:

$\forall x, y \in S$ such that $x \ne y$, both:
$\exists U \in \tau: x \in U, y \notin U$
and:
$\exists V \in \tau: y \in V, x \notin V$

### Definition 2

$\left({S, \tau}\right)$ is a Fréchet space or $T_1$ space if and only if all points of $S$ are closed in $T$.

## Also known as

A Fréchet Space is also commonly referred to as a $T_1$ space. On $\mathsf{Pr} \infty \mathsf{fWiki}$ both terms are used, frequently together.

A $T_1$ space is also known as an accessible space.

## Also see

• Results about $T_1$ (Fréchet) spaces can be found here.

Not to be confused with Definition:Fréchet Space (Functional Analysis).

## Source of Name

This entry was named for Maurice René Fréchet.