Exercise 11, § 1

Let X be an absolute neighbourhood retract.

(a) Prove that X is “semi-locally contractible”, that is to say that every point $x$ of X possesses a neighbourhood V such that the canonical injection of $(V, x)$ into X is strictly homotopic to the constant mapping of image $x$. (Use Exercise 9.)

(b) Let A be a compact topological space. Prove that the arcwise connected components of the space $\mathscr{C}_c(A; X)$ are open.

(c) Prove that X is locally arcwise connected.