Definition 9

The quotient topological space $Y/R$ is called the space obtained by attaching the space B to the space X along the mapping $f$. It is denoted by $X\cup_fB$.

Let $\alpha_f: X\rightarrow X\cup_fB$ and $\beta_f: B\rightarrow X\cup_fB$ denote the restrictions to $Y_1$ and $Y_2$ of the canonical surjection of Y onto $Y/R$. One has $\alpha_f\circ f=\beta_f|A$.