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https://math.stackexchange.com/questions/60590/category-theoretic-limit-related-to-topological-limit/62800#62800
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Let (X,O) be a topological space, F(X) the poset of filters on X with respect to inclusions, considered as a (small, thin) category in the usual way. Given x∈X and F∈F(X) let UX(x) denote the neighbourhood filter of x in (X,O) and Fx,F(X) the full subcategory of F(X) generated by {G∈F(X):F∪UX(x)⊆G}, let E:Fx,F↪F(X) be the obvious (embedding) diagram, Δ the usual diagonal functor and λ:Δ(F)→E the natural transformation where λ(G):F↪G is the inclusion for each G∈Fx,F. It is not hard to see that F tends to x in (X,O) iff λ is a limit of E. Kind regards - Stephan F. Kroneck.