Exercise 21, § 10

a) Let X be an accessible and locally quasi-compact space (I, p. 111, exerc. 30). Let $X_0$ be the discrete space having the same underlying set as X, and let $\tilde{X}_0$ be the (compact) space of ultrafilters on $X_0$ (the discrete space $X_0$ being identified with an open subspace of $\tilde{X}_0$) (I, p. 110, exerc. 27). In $\tilde{X}_0$, let R (or $R(X)$) be the equivalence relation between ultrafilters $\mathfrak{U}, \mathscr{V}$: “the sets of limit points of $\mathfrak{U}$ and of $\mathscr{V}$ in X are the same”. Now prove that the relation R is separated (use the description of the topology of $\tilde{X}_0$ given in I, p. 110, exerc. 26 b), exerc. 7 b) of I, p. 100 and axiom (C’)); the compact space $X^c = \tilde{X}_0 / R$ is in one-to-one correspondence with the set of primitive sets of X (I, p. 100, exerc.7). If X is separated (and consequently locally compact), $X^c$ identifies with X when X is compact, with the Alexandroff compactification of X when X is not compact.

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

c) Now prove 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 by this mapping of every compact subset of $X^c_*$ is a compact closed subset of X. (To see that $f$ is continuous, use b); observe moreover that the inverse image in $\tilde{X}_0$ of a point of $X^c_*$ is composed of ultrafilters on X having all a single limit point, and that if $\mathfrak{U}$ is an ultrafilter on X, its limit point in $\tilde{X}_0$, when $\mathfrak{U}$ is considered as a basis of ultrafilter on $\tilde{X}_0$, is $\mathfrak{U}$ considered as a point of $\tilde{X}_0$.)

d) Let Z be the sum set of a countable family of spaces $Y_n$ identical to the space XS defined in exerc. 7 of I, p.101 , and of a set $\{ \omega \}$ reduced to a point. A topology is defined on Z by taking as a fundamental system of neighbourhoods of a point $z \in Y_n$ the filter of its neighbourhoods in the space $Y_n$, and as a fundamental system of neighbourhoods of $\omega$ the set of $V_n$, where $V_n$ is the union of $\{ \omega \}$ and of the $Y_m$ of index $\geq n$. The space Z thus 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 a second accessible and locally quasi-compact space, and let u be a continuous mapping from X into Y, such that the inverse image by u of every quasi-compact subset of Y is quasi-compact. If one considers u as a mapping from $X^c_*$ into $Y^c_*$, now prove that u is continuous and extends (in a unique way) to a continuous mapping $X^c \to Y^c$. (First extend u to a continuous mapping from $\tilde{X}_0$ into $\tilde{Y}_0$ and prove that it is compatible with the equivalence relations $R(X)$ and $R(Y)$).

f) Let X be the accessible quasi-compact and not locally quasi-compact space defined in exerc. 30 c) of I, p. 111. Now prove that in the corresponding space $\tilde{X}_0$, the equivalence relation $R(X)$ is not separated.