Given two topological spaces \(X\) and \(Y\), a closed subset \(A\subseteq Y\) and a continuous map \(f\colon A\to X\). Let \(\sim\) be the equivalence relation on the disjoint union \(X \sqcup Y\) generated by \(a\sim f(a)\) for all \(a\in A\). We denote the resulting quotient space with
\[ X\cup_f Y:=X\sqcup Y/\sim. \]We call any such quotient space adjunction space and it is formed by attaching \(Y\) to \(X\) along \(f\).
Remarks
- The corresponding quotient map satisfies \(X\cup_f Y=q(X)\cup q(Y\setminus A)\). Furthermore, \(q(X)\) is closed and \(q(Y\setminus A)\) is open.