Editing Metric space (section)

===Compactness===
{{Main|Compact space}}
Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space.  There are several equivalent definitions of compactness in metric spaces:
# A metric space {{mvar|M}} is compact if every open cover has a finite subcover (the usual topological definition).
# A metric space {{mvar|M}} is compact if every sequence has a convergent subsequence.  (For general topological spaces this is called [[sequentially compact space|sequential compactness]] and is not equivalent to compactness.)
# A metric space {{mvar|M}} is compact if it is complete and totally bounded.  (This definition is written in terms of metric properties and does not make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.)
One example of a compact space is the closed interval {{closed-closed|0, 1}}.

Compactness is important for similar reasons to completeness: it makes it easy to find limits.  Another important tool is [[Lebesgue's number lemma]], which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover.
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Every compact metric space is [[second countable]], and is a continuous image of the [[Cantor set]]. (The latter result is due to [[Pavel Alexandrov]] and [[Pavel Samuilovich Urysohn|Urysohn]].)

===Locally compact and proper spaces===
A space is said to be ''[[locally compact]]'' if every point has a compact neighborhood. Euclidean spaces are locally compact, but infinite-dimensional [[Banach space]]s are not.

A metric space is [[proper space|''proper'']] if every ''closed ball'' <math>\{y\, \colon d(x,y)\leq r\}</math> is compact. A space is proper if and only if it is complete and locally compact.-->