Editing Continuous function (section)

===Defining topologies via continuous functions===
Given a function
<math display="block">f : X \to S,</math>
where ''X'' is a topological space and ''S'' is a set (without a specified topology), the [[final topology]] on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which <math>f^{-1}(A)</math> is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is [[Comparison of topologies|coarser]] than the final topology on ''S''. Thus, the final topology is the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is [[surjective]], this topology is canonically identified with the [[quotient topology]] under the [[equivalence relation]] defined by ''f''.

Dually, for a function ''f'' from a set ''S'' to a topological space ''X'', the [[initial topology]] on ''S'' is defined by designating as an open set every subset ''A'' of ''S'' such that <math>A = f^{-1}(U)</math> for some open subset ''U'' of ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus, the initial topology is the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the [[subspace topology]] of ''S'', viewed as a subset of ''X''.

A topology on a set ''S'' is uniquely determined by the class of all continuous functions <math>S \to X</math> into all topological spaces ''X''. [[Duality (mathematics)|Dually]], a similar idea can be applied to maps <math>X \to S.</math>