# Definition:Fréchet Space (Topology)

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*This page is about Fréchet Space in the context of topology. For other uses, see Fréchet Space.*

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

### Definition 1

$\struct {S, \tau}$ 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

$\struct {S, \tau}$ 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**.

## 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):**T-axioms**or**Tychonoff conditions**:**1.**(**$T_1$-space**or**Fréchet Space**)