Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪_f Y (sometimes also written as X +_f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally,

${\displaystyle X\cup _{f}Y=(X\sqcup Y)/\sim }$

where the equivalence relation ~ is generated by a ~ f(a) for all a in A, and the quotient is given the quotient topology. As a set, X ∪_f Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction.

Intuitively, one may think of Y as being glued onto X via the map f.