Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological spaces, and let ${\displaystyle A}$ be a subspace of ${\displaystyle Y}$. Let ${\displaystyle f:A\rightarrow X}$ be a continuous map (called the attaching map). One forms the adjunction space ${\displaystyle X\cup _{f}Y}$ (sometimes also written as ${\displaystyle X+_{f}Y}$) by taking the disjoint union of ${\displaystyle X}$ and ${\displaystyle Y}$ and identifying ${\displaystyle a}$ with ${\displaystyle f(a)}$ for all ${\displaystyle a}$ in ${\displaystyle A}$. Formally,

${\displaystyle X\cup _{f}Y=(X\sqcup Y)/\sim }$

where the equivalence relation ${\displaystyle \sim }$ is generated by ${\displaystyle a\sim f(a)}$ for all ${\displaystyle a}$ in ${\displaystyle A}$, and the quotient is given the quotient topology. As a set, ${\displaystyle X\cup _{f}Y}$ consists of the disjoint union of ${\displaystyle X}$ and (${\displaystyle Y-A}$). The topology, however, is specified by the quotient construction.

Intuitively, one may think of ${\displaystyle Y}$ as being glued onto ${\displaystyle X}$ via the map ${\displaystyle f}$.