Separation axioms

Fréchet space

A Fréchet space or T1 space is a topological space (𝑋,T) where any two points are separable, i.e. for any 𝑥,𝑦 ∈𝑋 with 𝑥 ≠𝑦 there exists an open neighbourhood 𝑈 of 𝑥 and 𝑉 of 𝑦 such that 𝑥 ∉𝑉 and 𝑦 ∉𝑈. topology Equivalent conditions include

1. for every 𝑥 ∈𝑋, the singleton {𝑥} is closed

Proof of equivalence

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

𝑋∖{𝑥}=⋃𝑦∈𝑋∖{𝑥}𝑈(𝑦)

is open, thus {𝑥} is closed.

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