---
tag: 662659CEK4
kind: definition
name: Adjunction space
msc: [54B17, 54B15]
defines: N6DQ3Q0HBT
uses: [0CQHFPGJ45, 0ZCK8DQYN1, 0VGFHTM75Y, ZXJKSB30WF, 0G8VEZ3WTK, 0NQ1H2X495]
sources:
  - url: https://en.wikipedia.org/wiki/Adjunction_space
---

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], let $A$ be a [[closed|0ZCK8DQYN1]] subset of $Y$, and let $f \colon A \to X$ be a [[continuous map|0VGFHTM75Y]].
Let $\sim$ be the smallest [[equivalence relation|0G8VEZ3WTK]] on the [[disjoint union|ZXJKSB30WF]] $X \sqcup Y$ such that $a \sim f(a)$ for every $a \in A$.
The *adjunction space* is the quotient set
$$X \cup_f Y = (X \sqcup Y) / {\sim},$$
with the [[quotient topology|0NQ1H2X495]].
The map $f$ is called the attaching map.

## Remarks

The composite $X \to X \sqcup Y \to X \cup_f Y$ is a closed [[topological embedding|00ZG9Y9TR4]], and $Y \setminus A$ maps homeomorphically onto an open subset.
Many authors only ask that $A$ be a subspace; the closedness of $A$ is needed for these two facts.

Example: let $Y = D^2$ be the closed unit disc, $A = S^1$ its boundary circle, and $X = S^1$ with $f(z) = z^2$.
Then $f$ is continuous and $A$ is closed in $Y$, so $X \cup_f D^2$ is an adjunction space.
It is homeomorphic to the real projective plane.

Example: if $X$ is a one-point space, then $X \cup_f Y$ is the quotient $Y / A$ that collapses $A$ to a point.

Non-example: let $Y = [0, 1]$, let $A = (0, 1]$, let $X$ be a one-point space, and let $f$ be the constant map.
The clause that $A$ is closed fails, and the quotient is a two-point space whose open sets are $\varnothing$, $\set{[A]}$ and the whole space.
There the image of $X$ is the open point $[A]$, which is not closed.