Exercise 4, § 5

Let $L$ be the Alexandroff half-line (TG, IV, p. 49, Exercise 12) and let $L^*$ be its Alexandroff compactification; let $\omega$ be the unique point of $L^*-$ L. Let A be a set having at least two elements and let $a$ be a point of A; put $X = L^*\times A$ and Y = X $-\{(\omega , a)\}$.

(a) Prove that the canonical injection $j$ of Y into X is étale, separated, and has the path lifting property.

(b) Prove that the mapping $j$ does not make Y a covering of X.