Editing Isolated point (section)

===Standard examples===

[[Topological space]]s in the following three examples are considered as [[subspace topology|subspaces]] of the [[real line]] with the standard topology.

* For the set <math>S=\{0\}\cup [1, 2],</math> the point 0 is an isolated point.
* For the set <math>S=\{0\}\cup \{1, \tfrac 1 2, \tfrac 1 3, \dots \},</math> each of the points {{tmath|\tfrac 1 k}} is an isolated point, but {{math|0}} is not an isolated point because there are other points in {{mvar|S}} as close to {{math|0}} as desired.
* The set <math>\N = \{0, 1, 2, \ldots \}</math> of [[natural number]]s is a discrete set.
In the topological space <math>X=\{a,b\}</math> with topology <math>\tau=\{\emptyset,\{a\},X\},</math> the element {{mvar|a}} is an isolated point, even though <math>b</math> belongs to the [[Closure (mathematics)|closure]] of <math>\{a\}</math> (and is therefore, in some sense, "close" to {{mvar|a}}). Such a situation is not possible in a [[Hausdorff space]].

The [[Morse theory#Morse lemma|Morse lemma]] states that [[non-degenerate critical point]]s of certain functions are isolated.