Editing General topology (section)

===Subspace and quotient===
Every subset of a topological space can be given the [[subspace topology]] in which the open sets are the intersections of the open sets of the larger space with the subset.  For any [[indexed family]] of topological spaces, the product can be given the [[product topology]], which is generated by the inverse images of open sets of the factors under the [[projection (mathematics)|projection]] mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

A [[Quotient space (topology)|quotient space]] is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a [[surjection|surjective]] [[function (mathematics)|function]], then the [[quotient topology]] on ''Y'' is the collection of subsets of ''Y'' that have open [[inverse image]]s under ''f''. In other words, the quotient topology is the finest topology on ''Y'' for which ''f'' is continuous.  A common example of a quotient topology is when an [[equivalence relation]] is defined on the topological space ''X''.  The map ''f'' is then the natural projection onto the set of [[equivalence class]]es.