Exercise 5, § 9

a) Show that the non-Hausdorff accessible space $Y = X/S$ defined in Exercise 7 of is such that every point of $Y$ has a compact neighbourhood; that the images $\alpha, \beta$ of 1 and — 1 in $Y$ have compact neighbourhoods which are not closed; that the intersection of a compact neighbourhood of $\alpha$ and a compact neighbourhood of $\beta$ is not quasi-compact, and that the union of these two neighbourhoods is not compact.

b) Let $X$ be the sum of $\mathbf{N}$ and an infinite set $A$. For each $x \in X$ let $\mathfrak{G}(x)$ be the filter base defined as follows: if $x \in \mathbf{N}$, then $\mathfrak{G}(x)$ consists of the single set $\{ x \}$; if $x \in A$ and if $S_{x, n}$ denotes the set whose elements are $x$ and the integers $\geq n$, then $\mathfrak{G}(x)$ consists of the sets $S_{x, n}$ where $n$ runs through $\mathbf{N}$. Show that there is a topology on $X$ for which $\mathfrak{G}(x)$ is a fundamental system of neighbourhoods of $x$ for every $x \in X$, and that $X$ is accessible but not Hausdorff in this topology. Show also that $X$ is not quasi-compact, but that $X$ has dense compact subsets.

c) Show that two topologies on a set $X$, with respect to each of which $X$ is accessible and quasi-compact, can be distinct but comparable [cf. § 8, Exercises 7 and 5 b)].