Exercise 6, § 1

Let X be a metrizable topological space of countable type. We say that X is an absolute retract [^1] if, for every metrizable topological space of countable type Y and every closed subset B of Y, every continuous mapping $f: B\rightarrow X$ extends to a continuous mapping of Y into X.

a) Let X be a metrizable topological space of countable type. In order that X be an absolute 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 exists a continuous mapping $g: Y\rightarrow X$ such that $g\circ f=$ Id$_X$. b) Exercise prove that the spaces $\mathbf{I}$ and $\mathbf{R}$ are absolute retracts.

c) Exercise prove that the product space of a countable family of absolute retracts is an absolute retract.

d) Let X be an absolute retract, let A be a subspace of X which is a retract of X. Exercise prove that A is an absolute retract.

[^1]: This exercise and the following ones reproduce the articles of K. Borsuk, « Sur les rétractes », Fund. Math. 17 (1931), p. 152–170 ; « Über eine Klasse von lokal zusammenhängenden Räumen », Fund. Math. 19 (1932), p. 220–242 ; O. Hanner, « Some theorems on absolute neighborhood retracts », Ark. Mat. 1, (1951), p. 389–408.