Editing Arzelà–Ascoli theorem (section)

===Functions on non-compact spaces===
The Arzela-Ascoli theorem generalises to functions <math>X \rightarrow Y</math> where <math>X</math> is not compact. Particularly important are cases where <math>X</math> is a [[topological vector space]]. Recall that if <math>X</math>
is a [[topological space]] and <math>Y</math> is a [[uniform space]] (such as any metric space or any [[topological group]], metrisable or not), there is the [[topology of compact convergence]] on the set <math>\mathfrak{F}(X,Y)</math> of functions <math>X \rightarrow Y</math>; it is set up so that a sequence (or more generally a 
[[Filter (set theory)|filter]] or [[Net (mathematics)|net]]) of functions converges if and only if it converges ''uniformly'' on each compact subset of <math>X</math>. Let <math>\mathcal{C}_c(X,Y)</math> be the subspace of
<math>\mathfrak{F}(X,Y)</math> consisting of continuous functions, equipped with the topology of compact convergence.
Then one form of the Arzelà-Ascoli theorem is the following:

:Let <math>X</math> be a topological space, <math>Y</math> a [[Hausdorff space|Hausdorff]] uniform space and <math>H\subset\mathcal{C}_c(X,Y)</math> an [[equicontinuous]] set of continuous functions such that <math>\{h(x) : h \in H\}</math> is [[Relatively compact subspace|relatively compact]] in <math>Y</math> for each <math>x\in X</math>. Then <math>H</math> is relatively compact in <math>\mathcal{C}_c(X,Y)</math>.

This theorem immediately gives the more specialised statements above in cases where <math>X</math> is compact
and the uniform structure of <math>Y</math> is given by a metric. There are a few other variants in terms of
the topology of [[precompact space|precompact]] convergence or other related topologies on 
<math>\mathfrak{F}(X,Y)</math>. It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of <math>X</math> by compact subsets. For details one can consult Bourbaki (1998), Chapter X, § 2, nr 5.