Editing Arzelà–Ascoli theorem (section)

=== Compact metric spaces and compact Hausdorff spaces ===
The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary compact [[metric space]]s and, more generally still, [[compact set|compact]] [[Hausdorff space]]s.  Let ''X'' be a compact Hausdorff space, and let ''C''(''X'') be the space of real-valued [[continuous function]]s on ''X''. A subset {{math|'''F''' ⊂ ''C''(''X'')}} is said to be ''equicontinuous'' if for every ''x''&nbsp;∈ ''X'' and every {{math|''ε'' > 0}}, ''x'' has a neighborhood ''U<sub>x</sub>'' such that

:<math>\forall y \in U_x, \forall f \in \mathbf{F} : \qquad |f(y) - f(x)| < \varepsilon.</math>

A set {{math|'''F''' ⊂ ''C''(''X'', '''R''')}} is said to be ''pointwise bounded'' if for every ''x''&nbsp;∈ ''X'',

:<math>\sup \{ | f(x) | : f \in \mathbf{F} \} < \infty.</math>

A version of the Theorem holds also in the space ''C''(''X'') of real-valued continuous functions on a [[compact set|compact]] [[Hausdorff space]] ''X'' {{harv|Dunford|Schwartz|1958|loc=§IV.6.7}}:

:Let ''X'' be a compact Hausdorff space. Then a subset '''F''' of ''C''(''X'') is [[relatively compact]] in the topology induced by the [[uniform norm]] [[if and only if]] it is [[equicontinuous]] and pointwise bounded.

The Arzelà–Ascoli theorem is thus a fundamental result in the study of the algebra of [[continuous functions on a compact Hausdorff space]].

Various generalizations of the above quoted result are possible.  For instance, the functions can assume values in a metric space or (Hausdorff) [[topological vector space]] with only minimal changes to the statement (see, for instance, {{harvtxt|Kelley|Namioka|1982|loc=§8}}, {{harvtxt|Kelley|1991|loc=Chapter 7}}):

:Let ''X'' be a compact Hausdorff space and ''Y'' a metric space. Then {{math|'''F''' ⊂ ''C''(''X'', ''Y'')}} is compact in the [[compact-open topology]] if and only if it is [[equicontinuous]], pointwise [[relatively compact]] and closed.

Here pointwise relatively compact means that for each ''x''&nbsp;∈&nbsp;''X'', the set {{math|'''F'''<sub>''x''</sub> {{=}} {&thinsp;''f''&thinsp;(''x'') : &thinsp;''f''&thinsp; ∈ '''F'''} }}is relatively compact in ''Y''.

In the case that ''Y'' is [[complete metric space|complete]], the proof given above can be generalized in a way that does not rely on the [[separable metric space|separability]] of the domain.  On a [[compact Hausdorff space]] ''X'', for instance, the equicontinuity is used to extract, for each ε&nbsp;=&nbsp;1/''n'', a finite open covering of ''X'' such that the [[oscillation (mathematics)|oscillation]] of any function in the family is less than ε on each [[open set]] in the cover.  The role of the rationals can then be played by a set of points drawn from each open set in each of the countably many covers obtained in this way, and the main part of the proof proceeds exactly as above. A similar argument is used as a part of the proof for the general version which does not assume completeness of ''Y''.