Definition:Fréchet Space (Topology)

Definition

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

$\left({X, \vartheta}\right)$ is a Fréchet space or $T_1$ space iff:

$\forall x, y \in X$ such that $x \ne y$, both:

$\exists U \in \vartheta: x \in U, y \notin U$

and:

$\exists V \in \vartheta: y \in V, x \notin V$

That is, for any two distinct points $x, y \in X$ there exist open sets $U, V \in \vartheta$ such that $x$ is in $U$ but not in $V$, and $y$ is in $V$ but not in $U$.

That is:

$\left({X, \vartheta}\right)$ is $T_1$ when every two points in $X$ are separated.

Equivalent Definitions

$\left({X, \vartheta}\right)$ is a Fréchet space or $T_1$ space iff all points are closed.

This is proved in Equivalent Definitions for $T_1$ Space.

Variants of Name

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

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$