Definition:Fréchet Space (Topology)

Definition
A topological space $\left({X, \vartheta}\right)$ is $T_1$ when 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, if both of the following happen:
 * $\exists U \in \vartheta: x \in U, y \notin U$
 * $\exists V \in \vartheta: y \in V, x \notin V$

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

Such a space is called a Fréchet space.