Editing Isolated point (section)

== Related notions ==
Any discrete subset {{mvar|S}} of Euclidean space must be [[countable]], since the isolation of each of its points together with the fact that [[Rational number|rationals]] are [[dense set|dense]] in the [[real number|reals]] means that the points of {{mvar|S}} may be mapped injectively onto a set of points with rational coordinates, of which there are only countably many.  However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example. 

A set with no isolated point is said to be ''[[dense-in-itself]]'' (every neighbourhood of a point contains  other points of the set).  A [[closed set]] with no isolated point is called a ''[[perfect set]]'' (it contains all its limit points and no isolated points).

The number of isolated points is a [[topological invariant]], i.e. if two [[topological spaces]] {{mvar|X, Y}} are [[homeomorphic]], the number of isolated points in each is equal.