Definition:Fréchet Space (Topology)

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

• Equivalence of Definitions of $T_1$ Space
• Definition:Tychonoff Separation Axioms

• 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)