Editing Real analysis (section)

=== Compactness ===
{{Main|Compactness}}Compactness is a concept from [[general topology]] that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being ''closed'' and ''bounded''. (In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.)  Briefly, a [[closed set]] contains all of its [[Boundary (topology)|boundary points]], while a set is [[Bounded set|bounded]] if there exists a real number such that the distance between any two points of the set is less than that number.  In <math>\mathbb{R}</math>, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, [[Interval (mathematics)|closed intervals]], and their finite unions.  However, this list is not exhaustive; for instance, the set <math>\{1/n:n\in\mathbb{N}\}\cup \{0}\</math> is a compact set; the [[Cantor set|Cantor ternary set]] <math>\mathcal{C}\subset [0,1]</math> is another example of a compact set.  On the other hand, the set <math>\{1/n:n\in\mathbb{N}\}</math> is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set.  The set <math>[0,\infty)</math> is also not compact because it is closed but not bounded.

For subsets of the real numbers, there are several equivalent definitions of compactness.

'''Definition.''' A set <math>E\subset\mathbb{R}</math> is compact if it is closed and bounded.

This definition also holds for Euclidean space of any finite dimension, <math>\mathbb{R}^n</math>, but it is not valid for metric spaces in general. The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the [[Heine–Borel theorem|Heine-Borel theorem]].

A more general definition that applies to all metric spaces uses the notion of a subsequence (see above).

'''Definition.''' A set <math>E</math> in a metric space is compact if every sequence in <math>E</math> has a convergent subsequence.

This particular property is known as ''subsequential compactness''. In <math>\mathbb{R}</math>, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above.  Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.

The most general definition of compactness relies on the notion of ''open covers'' and ''subcovers'', which is applicable to topological spaces (and thus to metric spaces and <math>\mathbb{R}</math> as special cases).  In brief, a collection of open sets <math>U_{\alpha}</math> is said to be an ''open cover'' of set <math>X</math> if the union of these sets is a superset of <math>X</math>.  This open cover is said to have a ''finite subcover'' if a finite subcollection of the <math>U_{\alpha}</math> could be found that also covers <math>X</math>.

'''Definition.''' A set <math>X</math> in a topological space is compact if every open cover of <math>X</math> has a finite subcover.

Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent. As another example, the image of a compact metric space under a continuous map is also compact.