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

 $y\sim f(y)\quad\text{for all}\quad y\in A.$

$Z$ 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 $Y$.

Title adjunction space AdjunctionSpace 2013-03-22 13:25:56 2013-03-22 13:25:56 antonio (1116) antonio (1116) 10 antonio (1116) Definition msc 54B17 QuotientSpace adjunction