Editing General topology (section)

===Defining topologies via continuous functions===
Given a function
:<math>f\colon X \rightarrow 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 ''f''<sup>−1</sup>(''A'') 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 can be characterized as 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'' has a basis of open sets given by those sets of the form ''f''<sup>−1</sup>(''U'') where ''U''  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 finer than the initial topology on ''S''.  Thus the initial topology can be characterized as 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 \rightarrow X</math> into all topological spaces ''X''. [[Duality (mathematics)|Dually]], a similar idea can be applied to maps <math>X \rightarrow S.</math>