Exercise 28, § 5

Let X be a topological space. One says that a homotopy $\sigma : X\times \mathbf{I}\rightarrow X$ is an isotopy if, for every $t\in \mathbf{I}$, the mapping $x\mapsto \sigma (x, t)$ deduced from $\sigma$ by passing to the subspace is a homeomorphism of X onto itself.

a) Let $m$ be a natural number and let $f: X\rightarrow \mathbf{R}^n$ be a continuous mapping. Prove that there exists an isotopy whose origin is the identity mapping of $X\times \mathbf{R}^n$ and whose term is the mapping of $X\times \mathbf{R}^n$ into $X\times \mathbf{R}^n$ given by $(x, y)\mapsto (x, y-f(x))$.

b) Let A and B be subspaces of X; one says that A is isotopic to B if there exists an isotopy $\sigma : X\times \mathbf{I}\rightarrow X$ whose origin is the identity mapping of X and such that $f(A\times \{1\}) = B$.

Prove that the relation “A is isotopic to B” is an equivalence relation in the set of subspaces of X (resp. in the set of closed subspaces of X). c) Let $m$ and $n$ be natural numbers, let A be a closed subspace of $\mathbf{R}^m$ and let B be a closed subspace of $\mathbf{R}^n$. Suppose that A and B are homeomorphic. Prove that the subspaces $A\times \{0\}$ and $\{0\} \times B$ of $\mathbf{R}^{m+n}$ are isotopic.