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# adjunction space

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 $Y$.

## Mathematics Subject Classification

54B17*no label found*

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## Comments

## Two comments on adjunction space

1. I have seen in some books this definition with the assumption that A is closed subspace. (Walker: Stone-Cech compactification, p.267; Engelking: General Topology)

I wasn't able to find a reference, where this assumption is relaxed. (But I would say, whenever I've used this notion, the condition of closedness wasn't necessary.)

Do you know about a book where adjunction space is used without this assumption?

2. Perhaps it could be mentioned in the article, that the adjunction space is pushout in the category of topological spaces.