Proposition 10

Let X be a topological space and let A be a subset of $x$. Suppose that there exists a homotopy $\sigma : X\times \mathbf{I}\rightarrow X$ with origin Id$_X$, constant on $A\times \{1\}$ and such that $\sigma (A\times \mathbf{I})\subset A$. Then the canonical mapping $\rho$ of X into $X/A$ is a homotopy equivalence.

Let $f$ be the terminal mapping of $\sigma$. It is a continuous mapping of X into X, homotopic to Id$_X$. It is constant on A; there therefore exists a unique continuous mapping $g: X/A\rightarrow X$ such that $g\circ \rho =f$. On the other hand, since $\sigma (A\times \mathbf{I})\subset A$, there exists a mapping $\sigma '$ of $X/A\times \mathbf{I}$ into $X/A$ such that $\sigma '(\rho (x), t) =\rho (\sigma (x, t))$ for every $x\in X$ and every $t\in \mathbf{I}$. By I, p. 19, prop. 10, the mapping $\sigma '$ is continuous. It is a homotopy joining the identity mapping of $X/A$ to the mapping $\rho \circ g$. Consequently, the mappings $g$ and $\rho$ are homotopy inverses of one another.