Exercise 15, § 2

Let $X = [a, b[$ be the locally compact space defined in Chapter I, § 9, Exercise 12.

a) A subset of $X$ is relatively compact if and only if it is bounded. Hence show that $X$ is countably compact and therefore (Proposition 15) non-metrizable (observe that every countable subset of $X$ is bounded).

b) Show that every point of $X$ has a metrizable neighbourhood (use Proposition 16).

c) If $A$ and $B$ are two non-compact closed sets in $X$, their intersection is not empty (form an increasing sequence in which the even-numbered terms are points of $A$ and the odd-numbered terms are points of $B$). Deduce that every neighbourhood of a non-compact closed set is the complement of a relatively compact set. If $A$ is the set of non-isolated points of $X$, show that $A$ is closed and is not the intersection of any countable family of open sets.

d) Let $X'$ denote the interval $[a, b]$ endowed with the following topology: for each $x \in X$ the intervals $]y, x]$, where $y$ runs through the set of elements $< x$, form a fundamental system of neighbourhoods of $x$; a fundamental system of neighbourhoods of $b$ is formed by the sets $V_x$ where, for each $x \in X$, $V_x$ denotes the union of $\{ b \}$ and the set of points of $[x, b]$ which have a predecessor (i.e., which are isolated in $X$). Show that $X'$ is countably compact but not regular [cf. Exercise 14 g)] and that the subspace $X$ of $X'$ is countably compact but not closed [cf. Exercise 14 c)].