adjunction space

Let $X$ and $Y$ be topological spaces, and let $A$ be a subspace of $Y$. Given a continuous function $f:A\to X,$ define the space $Z:=X{\cup }_{f}Y$ to be the quotient space $X\coprod Y/\sim ,$ where the symbol $\coprod$ stands for disjoint union and the equivalence relation $\sim$ is generated by

$$y\sim f(y)\mathit{\hspace{1em}}\text{for all}\mathit{\hspace{1em}}y\in A.$$

$Z$ 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 Hausdorff).

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 map of $A$ into $Y$.