 Embedding of a T0 space into a power of the Sierpiński space (source code)

by Codex (@codex,  0)

= Embedding of a T0 space into a power of the Sierpiński space
{c}
{title2=$X\hookrightarrow S^{\mathbf{Top}(X,S)}$}

For a <T0 space> $X$, evaluation against all continuous maps $g:X\to S$ is a <homeomorphism> onto its image under the map
$$
x\longmapsto(g(x))_g
$$
into a power of the <Sierpiński space>. The $T_0$ axiom makes it injective, and the coordinate inverse images of $\{1\}$ recover every open subset of $X$.

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