Editing Extreme value theorem (section)

==Generalization to metric and topological spaces==

When moving from the real line <math>\mathbb{R}</math> to [[metric spaces]] and general [[topological spaces]], the appropriate generalization of a closed bounded interval is a [[compact space|compact set]]. A set <math>K</math> is said to be compact if it has the following property: from every collection of [[open set]]s <math>U_\alpha</math> such that  <math display="inline">\bigcup U_\alpha \supset K</math>, a finite subcollection <math>U_{\alpha_1},\ldots,U_{\alpha_n}</math>can be chosen such that <math display="inline">\bigcup_{i=1}^n U_{\alpha_i} \supset K</math>. This is usually stated in short as "every open cover of <math>K</math> has a finite subcover". The [[Heine–Borel theorem]] asserts that a subset of the real line is compact if and only if it is both closed and bounded.  Correspondingly, a metric space has the [[Heine–Borel property]] if every closed and bounded set is also   compact.

The concept of a continuous function can likewise be generalized.  Given topological spaces <math>V,\ W</math>, a function <math>f:V\to W</math> is said to be continuous if for every open set <math>U\subset W</math>, <math>f^{-1}(U)\subset V</math> is also open.  Given these definitions, continuous functions can be shown to preserve compactness:<ref name=":0">{{Cite book|url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw Hill|year=1976|isbn=0-07-054235-X|location=New York|pages=89–90}}</ref>

{{Math theorem|If <math>V,\ W</math> are topological spaces, <math>f:V\to W</math> is a continuous function, and <math>K\subset V</math> is compact, then <math>f(K)\subset W</math> is also compact.}}

In particular, if <math>W = \mathbb{R}</math>, then this theorem implies that <math>f(K)</math> is closed and bounded for any compact set <math>K</math>, which in turn implies that <math>f</math> attains its [[Infimum and supremum|supremum]] and [[Infimum and supremum|infimum]] on any (nonempty) compact set <math>K</math>.  Thus, we have the following generalization of the extreme value theorem:<ref name=":0" />

{{Math theorem|If <math>K</math> is a nonempty compact set and <math>f:K\to \mathbb{R}</math> is a continuous function, then <math>f</math> is bounded and there exist <math>p,q\in K</math> such that <math>f(p)=\sup_{x\in K} f(x)</math> and <math>f(q) = \inf_{x\in K} f(x)</math>.
}}

Slightly more generally, this is also true for an upper semicontinuous function. (see [[compact space#Functions and compact spaces]]).