극한
Let [math]\displaystyle{ (X,\mathcal O) }[/math] be a topological space, [math]\displaystyle{ \mathcal F(X) }[/math] the partialy ordered set of filters on [math]\displaystyle{ X }[/math] with respect to inclusions, considered as a small category in the usual way. Given [math]\displaystyle{ x\in X }[/math] and [math]\displaystyle{ F\in\mathcal F(X) }[/math] let [math]\displaystyle{ \mathcal U_X(x) }[/math] denote the neighbourhood filter of [math]\displaystyle{ x }[/math] in [math]\displaystyle{ (X,\mathcal O) }[/math] and [math]\displaystyle{ \mathcal F_{x,F}(X) }[/math] the full subcategory of [math]\displaystyle{ \mathcal F(X) }[/math] generated by [math]\displaystyle{ \{G\in\mathcal F(X):F\cup\mathcal U_X(x)\subseteq G\} }[/math], let [math]\displaystyle{ E:\mathcal F_{x,F}\hookrightarrow\mathcal F(X) }[/math] be the obvious (embedding) diagram, [math]\displaystyle{ \Delta }[/math] the usual diagonal functor and [math]\displaystyle{ \lambda:\Delta(F)\rightarrow E }[/math] the natural transformation where [math]\displaystyle{ \lambda(G):F\hookrightarrow G }[/math] is the inclusion for each [math]\displaystyle{ G\in\mathcal F_{x,F} }[/math]. It is not hard to see that [math]\displaystyle{ F }[/math] tends to [math]\displaystyle{ x }[/math] in [math]\displaystyle{ (X,\mathcal O) }[/math] iff [math]\displaystyle{ \lambda }[/math] is a limit of [math]\displaystyle{ E }[/math]. [1]