Proposition 8

Let Z be a topological space.

a) Let $u: X\rightarrow Z$ and $v: B\rightarrow Z$ be continuous mappings such that $v(a) =u(f(a))$ for every $a\in A$. Then there exists a unique continuous mapping $w: X\cup_fB\rightarrow Z$ such that $w\circ \alpha_f=u$ and $w\circ \beta_f=v$.

b) Let $\sigma : X\times \mathbf{I}\rightarrow Z$ and $\tau : B\times \mathbf{I}\rightarrow Z$ be homotopies. Suppose that $\tau$ is fixed on A and that $\sigma (f(a), t) =\tau (a, t)$ for every $a\in A$ and every $t\in \mathbf{I}$. Then there exists a unique homotopy $\eta : (X\cup_fB)\times \mathbf{I}\rightarrow Z$ such that $\eta (\alpha_f(x), t) =\sigma (x, t)$ and $\eta (\beta_f(b), t) =\tau (b, t)$ for $x\in X,b\in B$ and $t\in \mathbf{I}$. This homotopy is fixed on $\beta_f(A)$.

The proposition results immediately from the definition of a quotient space and prop. 10 (I, p. 19).