Exercise 18, § 1

One says that a topological space X is locally equiconnected if the diagonal mapping of X into $X\times X$ is a cofibration.

a) Prove that the product of a finite family of locally equiconnected spaces is locally equiconnected.

Let X be a locally equiconnected space.

b) Prove that there exists a covering of X by a sequence $(U_n)_{n\in\mathbf{N}}$ of open subsets of X such that, for every $n$, the injection of $U_n$ into X is homotopic to a constant mapping.

c) Prove that X is separated.

d) Let $u: X\rightarrow \mathbf{I}$ be a continuous mapping and let U be the set of points $x\in X$ such that $u(x)>0$. Prove that the path-connected components of U are open. In particular, the path-connected components of X are open.

e) Let A be a closed subspace of X which is a retract of X. Prove that A is locally equiconnected and that the pair $(X,A)$ has the homotopy extension property. (“Dyer and Eilenberg Theorem”.)

f) Let A be a subspace of X such that the pair $(X,A)$ has the homotopy extension property. Prove that A is locally equiconnected.