^metadata > [!definition]+ Definition. ([[adjunction space]]) > Let $X,Y$ be [[topological space|topological spaces]] with $A$ a [[subspace topology|subspace]] of $Y$. Let $f:A \to X$ be [[continuous]] (called the **attaching map**). The **adjunction space** is $X \cup_{f} Y = \left( X \sqcup Y \right) / \sim,$ > where $\sim$ is the [[equivalence relation]] generated by $a \sim f(a)$. > Here, $X \sqcup Y$ is given the [[disjoint union topology]] and the [[quotient space|quotient]] is given the [[quotient topology]]. ^definition > [!basicexample]+ > - The notion of [[cell complex|attaching an n-cell]] is obtained when setting $Y=\mathbb{D}^{n}$ ([[closed set|closed]]) and $S=\text{bd } Y= \mathbb{S}^{n-1}$; > - Suppose $X$ and $Y$ are [[pointed set|based]] at $x \in X$ and $y \in Y$. Then with $A:= \{ y \} \subset Y$ the attachment map $f:A \to X$ given by $f(y):=x$ recovers the notion of [[wedge sum]] $X \vee Y$. ^basic-example-1 > [!backlink]+ > >```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag >``` ^backlink > [!frontlink]+ >```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag >``` ^frontlink #reformatrevisebatch02