Definition:Fréchet Space (Topology)

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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.


Sources