Editing Subspace topology (section)

== Examples ==
In the following, <math>\mathbb{R}</math> represents the [[real number]]s with their usual topology.
* The subspace topology of the [[natural number]]s, as a subspace of <math>\mathbb{R}</math>, is the [[discrete topology]].
* The [[rational number]]s <math>\mathbb{Q}</math> considered as a subspace of <math>\mathbb{R}</math> do not have the discrete topology ({0} for example is not an open set in <math>\mathbb{Q}</math> because there is no open subset of <math>\mathbb{R}</math> whose intersection with <math>\mathbb{Q}</math> can result in ''only'' the [[Singleton (mathematics)|singleton]] {0}). If ''a'' and ''b'' are rational, then the intervals (''a'', ''b'') and [''a'', ''b''] are respectively open and closed, but if ''a'' and ''b'' are irrational, then the set of all rational ''x'' with ''a'' < ''x'' < ''b'' is both open and closed.
* The set [0,1] as a subspace of <math>\mathbb{R}</math> is both open and closed, whereas as a subset of <math>\mathbb{R}</math> it is only closed.
* As a subspace of <math>\mathbb{R}</math>, [0, 1] &cup; [2, 3] is composed of two disjoint ''open'' subsets (which happen also to be closed), and is therefore a [[disconnected space]].
* Let ''S'' = [0, 1) be a subspace of the real line <math>\mathbb{R}</math>. Then [0, {{frac|1|2}}) is open in ''S'' but not in <math>\mathbb{R}</math> (as for example the intersection between (-{{frac|1|2}}, {{frac|1|2}}) and ''S'' results in [0, {{frac|1|2}})). Likewise [{{frac|1|2}}, 1) is closed in ''S'' but not in <math>\mathbb{R}</math> (as there is no open subset of <math>\mathbb{R}</math> that can intersect with [0, 1) to result in [{{frac|1|2}}, 1)). ''S'' is both open and closed as a subset of itself but not as a subset of <math>\mathbb{R}</math>.