Is there any connection between category-theoretic term ‘limit’ (=universal cone) over diagram, and topological term ‘limit point’ of a sequence, function, net…?

To be more precise, is there a category-theoretic setting of some non-trivial topological space such that these different concepts of term ‘limit’ somehow relate?

This question came to me after I saw ( http://www.youtube.com/watch?v=be7rx29eMr4 ) a surprising fact that generalised metric spaces can be seen as categories enriched over preorder ([0,∞],≤).

**Answer**

The connection is well-known (in particular I’m claiming no originality; I don’t recall where I found this, though !): 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.

**Attribution***Source : Link , Question Author : Rafael Mrđen , Answer Author : bonnbaki*