\begin{definition}[Definition:Fréchet Space (Topology)/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ 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:
:$\struct {S, \tau}$ is $T_1$ {{iff}} every two elements of $S$ are separated.
\end{definition}