Editing Metric space (section)

===The topology of a metric space===
For any point {{mvar|x}} in a metric space {{mvar|M}} and any real number {{math|''r'' > 0}}, the [[ball (mathematics)|''open ball'']] of radius {{mvar|r}} around {{mvar|x}} is defined to be the set of points that are strictly less than distance {{mvar|r}} from {{mvar|x}}:
<math display="block">B_r(x)=\{y \in M : d(x,y) < r\}.</math>
This is a natural way to define a set of points that are relatively close to {{mvar|x}}.  Therefore, a set <math>N \subseteq M</math> is a [[neighborhood (mathematics)|''neighborhood'']] of {{mvar|x}} (informally, it contains all points "close enough" to {{mvar|x}}) if it contains an open ball of radius {{mvar|r}} around {{mvar|x}} for some {{math|''r'' > 0}}.

An ''open set'' is a set which is a neighborhood of all its points.  It follows that the open balls form a [[base (topology)|base]] for a topology on {{mvar|M}}.  In other words, the open sets of {{mvar|M}} are exactly the unions of open balls.  As in any topology, [[closed set]]s are the complements of open sets.  Sets may be both open and closed as well as neither open nor closed.

This topology does not carry all the information about the metric space.  For example, the distances {{math|''d''<sub>1</sub>}}, {{math|''d''<sub>2</sub>}}, and {{math|''d''<sub>∞</sub>}} defined above all induce the same topology on <math>\R^2</math>, although they behave differently in many respects.  Similarly, <math>\R</math> with the Euclidean metric and its subspace the interval {{open-open|0, 1}} with the induced metric are [[homeomorphism|homeomorphic]] but have very different metric properties.

Conversely, not every topological space can be given a metric.  Topological spaces which are compatible with a metric are called [[metrizable space|''metrizable'']] and are particularly well-behaved in many ways: in particular, they are [[paracompact space|paracompact]]<ref>Rudin, Mary Ellen.  [https://www.jstor.org/stable/2035708 A new proof that metric spaces are paracompact] {{webarchive|url=https://web.archive.org/web/20160412015215/http://www.jstor.org/stable/2035708 |date=2016-04-12 }}.  Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603.</ref> [[Hausdorff space]]s (hence [[normal space|normal]]) and [[first-countable space|first-countable]].{{efn|Balls with rational radius around a point {{mvar|x}} form a [[neighborhood basis]] for that point.}}  The [[Nagata–Smirnov metrization theorem]] gives a characterization of metrizability in terms of other topological properties, without reference to metrics.