# Definition:Fréchet Space (Topology)

Jump to navigation
Jump to search

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

## Contents

## 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.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**T-axioms**or**Tychonoff conditions**:**1.**(**$T_1$-space**or**Fréchet Space**)