Exercise 21, § 10

(*) a) Let $X$ be an accessible space which is locally quasi-compact (\S 9, Exercise 29). Let $X_0$ be the discrete space which has the same underlying set as $X$, and let $\tilde{X}_0$ be the (compact) ultrafilter space of $X_0$ (the discrete space $X_0$ being identified with an open subspace of $\tilde{X}_0$) (\S 9, Exercise 26). On $\tilde{X}_0$, let $R$ [or $R(X)$] be the following equivalence relation between ultrafilters $\mathfrak{U}, \mathfrak{B}$: "the sets of limit points of $\mathfrak{U}$ and $\mathfrak{B}$ in $X$ are the same". Show that $R$ is Hausdorff [use the description of the topology of $\tilde{X}_0$ given in \S 9, Exercise 25 b), Exercise 7 b) of \S 7 and axiom (C')]. The compact space $X^c = \tilde{X}_0 / R$ is in one-to-one correspondence with the set of primitive subsets of $X$ (\S 7, Exercise 7). If $X$ is Hausdorff (and therefore locally compact) then $X^c$ can be identified with $X$ if $X$ is compact, and with the one-point compactification of $X$ if $X$ is not compact

b) Let $Y$ be a closed subspace of $X$. The ultrafilter space $\tilde{Y}_0$ can be identified with the closure of $Y_0$ in $\tilde{X}_0$, and $Y^c$ with the canonical image of $\tilde{Y}_0$ in $X^c$.

c) Show that the restriction to $X_0$ of the canonical mapping $\tilde{X}_0 \to X^c$ is a bijection of $X_0$ onto a subspace $X_*^c$ of $X^c$, that the canonical mapping $f : X_*^c \to X$ is a continuous bijection, and that the image under this mapping of any compact subset of $X_*^c$ is a closed compact subset of $X$. [To show that $f$ is continuous, use b); also remark that the inverse image in $\tilde{X}_0$ of a point of $X_*^c$ consists of all the ultrafilters on $X$ which have a single given limit point, and that if $\mathfrak{U}$ is an ultrafilter on $X$, then its limit point in $\tilde{X}_0$, when $\mathfrak{U}$ is regarded as an ultrafilter base on $\tilde{X}_0$, is $\mathfrak{U}$ regarded as a point of $\tilde{X}_0$].

d) Let $Z$ be the set which is the sum of a countable number of spaces $Y_n$ each identical with the space $Y$ defined in Exercise 7 of § 8 and a set $\{ \omega \}$ consisting of a single point. Define a topology on $Z$ by taking as a fundamental system of neighbourhoods of a point $z \in Y_n$ its neighbourhood filter in the space $Y_n$, and as fundamental system of neighbourhoods of $\omega$ the set of subsets $V_n$ of $Y$, where $V_n$ is the union of $\{ \omega \}$ and the $Y_m$ where $m \geq n$. The space $Z$ so defined is accessible, quasi-compact and locally quasi-compact; but the subspace $Z_*^c$ of $Z^c$ is not locally compact.*

e) Let $Y$ be another locally quasi-compact accessible space, and let $u$ be a continuous mapping of $X$ into $Y$, such that the inverse image under $u$ of each quasi-compact subset of $Y$ is quasi-compact. Regarding $u$ as a mapping of $X_*^c$ into $Y_*^c$, show that $u$ is continuous and can be

(*) The results in this exercise and in Exercise 7 of § 7 are due to J. Fell.

(unicely) extended to a continuous mapping $X^c \to Y^c$. [First extend $u$ to a continuous mapping of $\tilde{X}_0$ into $\tilde{Y}_0$ and show that the extension is compatible with the equivalence relations $R(X),\ R(Y)$].

f) Let $X$ be the quasi-compact, not locally quasi-compact, accessible space defined in Exercise 29 c) of § 9. Show that in the corresponding space $\tilde{X}_0$ the equivalence relation $R(X)$ is not Hausdorff.