Editing Subspace topology (section)

== Definition ==

Given a topological space <math>(X, \tau)</math> and a [[subset]] <math>S</math> of <math>X</math>, the '''subspace topology''' on <math>S</math> is defined by
:<math>\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.</math>
That is, a subset of <math>S</math> is open in the subspace topology [[if and only if]] it is the [[intersection (set theory)|intersection]] of <math>S</math> with an [[open set]] in <math>(X, \tau)</math>. If <math>S</math> is equipped with the subspace topology then it is a topological space in its own right, and is called a '''subspace''' of <math>(X, \tau)</math>. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset <math>S</math> of <math>X</math> as the [[coarsest topology]] for which the [[inclusion map]]
:<math>\iota: S \hookrightarrow X</math>
is [[continuous (topology)|continuous]].

More generally, suppose  <math>\iota</math> is an [[Injective function|injection]] from a set <math>S</math> to a topological space <math>X</math>. Then the subspace topology on <math>S</math> is defined as the coarsest topology for which <math>\iota</math> is continuous. The open sets in this topology are precisely the ones of the form <math>\iota^{-1}(U)</math> for <math>U</math> open in <math>X</math>. <math>S</math> is then [[homeomorphic]] to its image in <math>X</math> (also with the subspace topology) and <math>\iota</math> is called a [[topological embedding]].

A subspace <math>S</math> is called an '''open subspace''' if the injection <math>\iota</math> is an [[open map]], i.e., if the forward image of an open set of <math>S</math> is open in <math>X</math>. Likewise it is called a '''closed subspace''' if the injection <math>\iota</math> is a [[closed map]].