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The nlab says the compact objects in the category of opens sets of a given space are precisely the compact sets.

I'm having trouble with this. I need to show equivalence between

  • $\varinjlim _\alpha \mathsf{Hom}(A,U_\alpha) = \mathsf{Hom}(A,X)$ for every filtered colimit
  • Each open cover of $A$ has an open subcover.

But I think $\varinjlim _\alpha \mathsf{Hom}(A,U_\alpha)$ is just $\coprod \mathsf{Hom}(A,U_\alpha)$ modulo functions corestrictions and I really don't know how to proceed.

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Colimits in the category of open subsets (and inclusions) are just unions. A filtered subcategory of the category of open subsets has the property that any finite union of objects is contained in another object.

So if $A$ is a compact open set then if there's a morphism to the union of the open subsets in a filtered subcategory (i.e., if $A$ is a subset of that union) then $A$ must be a subset of an object of that filtered subcategory.

But if $A$ is a non-compact open set and $\mathcal{O}$ is an open cover of $A$ without a finite subcover, then there is a morphism from $A$ to the union of $\mathcal{O}$ that doesn't factor through any object in the filtered subcategory consisting of finite unions of sets from $\mathcal{O}$.

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