Exercise 7, § 1

Let X be a topological space of countable type which is metrizable. We say that X is an absolute neighborhood retract if, for every topological space of countable type which is metrizable Y and every closed subset B of Y, every continuous mapping $f: B\rightarrow X$ extends to a continuous mapping of a neighborhood of B into X.

a) Let X be a topological space of countable type which is metrizable. In order that X be an absolute neighborhood retract, it is necessary and sufficient that, for every metrizable space of countable type Y and every continuous, injective and closed mapping $f$ of X into Y, there exist a neighborhood U of $f(X)$ and a continuous mapping $g: U\rightarrow X$ such that $g\circ f=$ Id$_X$.

b) Let X be an absolute neighborhood retract and let A be a subset of X. If A has a neighborhood U of which it is a retract, then A is an absolute neighborhood retract. In particular, every open subset of X, every retract of X, is an absolute neighborhood retract. c) Prove that the product space of a finite family of absolute neighborhood retracts is an absolute neighborhood retract.

d) Prove that the sphere $\mathbf{S}_n$ is an absolute neighborhood retract.

e) Let X be a topological space of countable type which is metrizable, let $A_1$ and $A_2$ be closed subsets of X such that $X = A_1\cup A_2$. Suppose that $A_1\cap A_2$ is an absolute neighborhood retract. In order that X be an absolute neighborhood retract, it is necessary and sufficient that the same be true of $A_1$ and $A_2$.

f) Let X be a topological space of countable type which is metrizable. If X is the union of two open subsets which are absolute neighborhood retracts, then X is an absolute neighborhood retract.

g) Let X be a topological space of countable type which is metrizable and such that every point has a neighborhood which is an absolute neighborhood retract. Prove that X is an absolute neighborhood retract.