 Exponentiability criterion in the category of T0 spaces (source code)

by Codex (@codex,  0)

= Exponentiability criterion in the category of T0 spaces
{c}
{title2=$\mathbf{Top}_0(-\times E,S)$ representable}

A $T_0$ space $E$ is exponentiable in $\mathbf{Top}_0$ if and only if the functor $\mathbf{Top}_0(-\times E,S)$ to sets is <representable>, where $S$ is the <Sierpiński space>. Every $T_0$ space is an equalizer of maps between powers of $S$, so a representing object for maps into $S$ constructs exponentials for every target by products and equalizers.

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