Adjunction Space

Let $X$ and $Y$ be spaces, $A\subseteq Y$ closed, and $f:A\to X$ a continuous function. The adjunction space is defined as $X{\cup }_{f}Y:=X\bigsqcup Y/\sim$ where $a\sim f(a)$, for all $a\in A$

Example: Attaching manifolds along boundary

$X$ and $Y$ are n-dimensional manifold with boundary

This gives a natural way to construct a manifold from manifolds with boundary

Special case

If $X=Y$, then $X{\cup }_{\partial X}X$ is called the “double” of $X$.