Exercise 17, § 1

Soit X a topological space and let A be a subspace of X. Denote by $i: A\rightarrow X$ and $j:$ Cyl($i$)$\rightarrow X\times \mathbf{I}$ the canonical injections. a) Suppose that $X =\mathbf{R}$ and $A = ]0,1[$; prove that the mapping $j$ is not strict.

b) Suppose that every point of X has a countable basis of closed neighbourhoods. For the mapping $j$ to be strict, it is necessary and sufficient that A be closed. c) Suppose that A is the Alexandroff ray (TG, IV, p. 49, exerc. 12) and that X is its Alexandroff compactification. Now prove that the mapping $j$ is strict though A is not closed in X.