Exercise 5, § 9

a) The accessible non-separated space Y = X/S defined in Exercise 7 of I, p. 101 is such that every point of Y has a compact neighbourhood; the images $\alpha, \beta$ of 1 and −1 in Y have non-closed compact neighbourhoods in Y; the intersection of a compact neighbourhood of $\alpha$ and a compact neighbourhood of $\beta$ is not quasi-compact, and their union is not compact.

b) Let X be the sum of $\mathbf{N}$ and an infinite set A; for every $x \in X$, let $\mathfrak{G}(x)$ be the filter basis defined as follows: if $x \in \mathbf{N}$, $\mathfrak{G}(x)$ is formed by the single set $\{x\}$; if $x \in A$, and if $S_{x,n}$ denotes the set formed by $x$ and the integers $\geq n$, $\mathfrak{G}(x)$ is the set of the $S_{x,n}$, where $n$ runs through $\mathbf{N}$. Prove that there is a topology on X for which $\mathfrak{G}(x)$ is a fundamental system of neighbourhoods of $x$ for every $x \in X$; X is accessible but non-separated for this topology; X is not quasi-compact, but there exist in X everywhere dense compact sets.

c) Prove that on the same set X, two topologies of accessible quasi-compact space can be comparable and distinct (cf. I, p. 101, Exercise 7 and I, p. 100, Exercise 5 b)).