Let and be topological spaces, and let be a subspace of . Given a continuous function define the space to be the quotient space where the symbol stands for disjoint union and the equivalence relation is generated by
is called an adjunction of to along (or along , if the map is understood). This construction has the effect of gluing the subspace of to its image in under
Though the definition makes sense for arbitrary , it is usually assumed that is a closed subspace of . This results in better-behaved adjunction spaces (e.g., the quotient of by a non-closed set is never Hausdorff).
The adjunction space construction is a special case of the pushout in the category of topological spaces. The two maps being pushed out are and the inclusion map of into .
|Date of creation||2013-03-22 13:25:56|
|Last modified on||2013-03-22 13:25:56|
|Last modified by||antonio (1116)|