Definition 25.1:

Given a topological space $X$, a closed subset $A$ and a continuous map $A\longrightarrow B$ we define an equivalence relation on the disjoint sum (coproduct) $X\sqcup B$ as follows

$$b\sim x$$    if and only if $$\;\; x\in A$$    and $$\; f(x) = b.$$

Thus a point $x\in A$ is identified with its image $f(x) \in B$. There are no other identifications besides this. The quotient space under this equivalence relation is called the adjunction space or the space obtained by attaching $X$ to $B$ via the map $f$. The space is denoted by $X\sqcup_f B$. Thus

$$X\sqcup_f B = (X\sqcup B)/\sim$$

The situation may be pictorially described as

Figure: Adjunction Space

[width=1.0]GKSBook/fig15/fig15.eps