Definition:Fréchet Space (Topology)/Definition 1
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ 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:
- $\struct {S, \tau}$ 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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms