Fréchet space

A Fréchet space or $T 1$ space is a topological space $(𝑋, T)$ where any two points are separable, i.e. for any $𝑥, 𝑦 \in 𝑋$ with $𝑥 \neq 𝑦$ there exists an open neighbourhood $𝑈$ of $𝑥$ and $𝑉$ of $𝑦$ such that $𝑥 \notin 𝑉$ and $𝑦 \notin 𝑈$ . #m/def/topology Equivalent conditions include

for every $𝑥 \in 𝑋$ , the singleton ${𝑥}$ is closed

Proof of equivalence

Let $𝑥 \in 𝑋$ . For each $𝑦 \in 𝑋 ∖ {𝑥}$ , let $𝑈 (𝑦)$ be an open neighbourhood of $𝑦$ not containing $𝑥$ . Then

𝑋 ∖ {𝑥} = ⋃ 𝑦 \in 𝑋 ∖ {𝑥} 𝑈 (𝑦)

is open, thus ${𝑥}$ is closed.

#state/tidy | #lang/en | #SemBr