Definition:Fréchet Space (Topology)/Definition 1

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

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


 * $\forall x, y \in S$ such that $x \ne y$, both:
 * $\exists U \in \tau: x \in U, y \notin U$
 * and:
 * $\exists V \in \tau: y \in V, x \notin V$

That is:
 * for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ such that $x$ is in $U$ but not in $V$, and $y$ is in $V$ but not in $U$.

That is:
 * $\left({S, \tau}\right)$ is $T_1$ iff every two elements of $S$ are separated.

Also see

 * Equivalence of Definitions of $T_1$ Space