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:


 * $\forall x, y \in X$, 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 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.