Editing Arzelà–Ascoli theorem (section)

==Necessity==
Whereas most formulations of the Arzelà–Ascoli theorem assert sufficient conditions for a family of functions to be (relatively) compact in some topology, these conditions are typically also necessary.  For instance, if a set '''F''' is compact in ''C''(''X''), the Banach space of real-valued continuous functions on a compact Hausdorff space with respect to its uniform norm, then it is bounded in the uniform norm on ''C''(''X'') and in particular is pointwise bounded. Let ''N''(''ε'', ''U'') be the set of all functions in '''F''' whose [[oscillation (mathematics)|oscillation]] over an open subset ''U'' ⊂ ''X'' is less than ''ε'':

:<math>N(\varepsilon, U) = \{f \mid \operatorname{osc}_U f < \varepsilon\}.</math>

For a fixed ''x''&isin;''X'' and ''ε'', the sets ''N''(''ε'', ''U'') form an open covering of '''F''' as ''U'' varies over all open neighborhoods of ''x''.  Choosing a finite subcover then gives equicontinuity.