Editing Compact space (section)

==== Characterization by continuous functions ====

Let {{mvar|X}} be a topological space and {{math|C(''X'')}} the ring of real continuous functions on {{mvar|X}}. 
For each {{math|''p'' ∈ ''X''}}, the evaluation map <math>\operatorname{ev}_p\colon C(X)\to \mathbb{R}</math>
given by {{math|1=ev<sub>''p''</sub>(''f'') = ''f''(''p'')}} is a ring homomorphism. 
The [[kernel (algebra)|kernel]] of {{math|ev<sub>''p''</sub>}} is a [[maximal ideal]], since the [[residue field]] {{nowrap|{{math|C(''X'')/ker ev<sub>''p''</sub>}}}} is the field of real numbers, by the [[first isomorphism theorem]]. A topological space {{mvar|X}} is [[pseudocompact space|pseudocompact]] if and only if every maximal ideal in {{math|C(''X'')}} has residue field the real numbers. For [[completely regular space]]s, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.<ref>{{harvnb|Gillman|Jerison|1976|loc=§5.6}}</ref>  There are pseudocompact spaces that are not compact, though.

In general, for non-pseudocompact spaces there are always maximal ideals {{mvar|m}} in {{math|C(''X'')}} such that the residue field {{math|C(''X'')/''m''}} is a ([[non-archimedean field|non-Archimedean]]) [[hyperreal field]]. The framework of [[non-standard analysis]] allows for the following alternative characterization of compactness:<ref>{{harvnb|Robinson|1996|loc=Theorem&nbsp;4.1.13}}</ref> a topological space {{mvar|X}} is compact if and only if every point {{mvar|x}} of the natural extension {{math|''*X''}} is [[infinitesimal|infinitely close]] to a point {{math|''x''<sub>0</sub>}} of {{mvar|X}} (more precisely, {{mvar|x}} is contained in the [[monad (non-standard analysis)|monad]] of {{math|''x''<sub>0</sub>}}).