Remark

In other words, a contraction of X onto A is a homotopy $\sigma : X\times \mathbf{I}\rightarrow X$ having the following properties:

(i) $\sigma (x,0) =x$ for every $x\in X$;

(ii) $\sigma (x,1)\in A$ for every $x\in X$;

(iii) $\sigma (x, t) =x$ for every $x\in A$ and every $t\in \mathbf{I}$.

For there to exist a contraction of X onto A, it is necessary and sufficient that the canonical injection of A into X be a homotopy of the pair $(A,A)$ onto the pair $(X,A)$.

Let $a$ be a point of X. A contraction of X onto the subspace $\{a\}$ is nothing other than a homotopy pointed at $a$ joining the mapping Id$_X$ to the constant mapping with image $\{a\}$. Consequently, X contracts onto $\{a\}$ if and only if X is contractible at $a$ (Example 3 of III, p. 234).

Let X be a topological space and let A be a subset of X. Let $\sigma$ be a contraction of X onto A. The mapping $r$ of X into A defined by $r(x) =\sigma (x,1)$ is a retraction of X onto A and $\sigma$ is a homotopy joining Id$_X$ to $r$. The relations $r\circ i=$ Id$_A$ and $i\circ r=r$ then imply that the mappings $i$ and $r$ are homotopies, inverse to one another up to homotopy.