Definition:Fréchet Space (Topology)/Definition 1

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Definition

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


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

$\forall x, y \in S$ 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 $T_1$ if and only if every two elements of $S$ are separated.


Also see

  • Results about $T_1$ (Fréchet) spaces can be found here.


Source of Name

This entry was named for Maurice René Fréchet.


Sources