adjunction space

Let X and Y be topological spacesMathworldPlanetmath, and let A be a subspaceMathworldPlanetmath of Y. Given a continuous functionMathworldPlanetmathPlanetmath f:AX, define the space Z:=XfY to be the quotient spaceMathworldPlanetmath XY/, where the symbol stands for disjoint unionMathworldPlanetmath and the equivalence relationMathworldPlanetmath is generated by

yf(y)for allyA.

Z is called an adjunction of Y to X along f (or along A, if the map f is understood). This construction has the effect of gluing the subspace A of Y to its image in X under f.

Remark 1

Though the definition makes sense for arbitrary A, it is usually assumed that A is a closed subspace of Y. This results in better-behaved adjunction spaces (e.g., the quotient of X by a non-closed set is never HausdorffPlanetmathPlanetmath).

Remark 2

The adjunction space construction is a special case of the pushout in the category of topological spaces. The two maps being pushed out are f and the inclusion mapMathworldPlanetmath of A into Y.

Title adjunction space
Canonical name AdjunctionSpace
Date of creation 2013-03-22 13:25:56
Last modified on 2013-03-22 13:25:56
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 10
Author antonio (1116)
Entry type Definition
Classification msc 54B17
Related topic QuotientSpace
Defines adjunction