Exercise 22, § 6

Let $X, Y$ be two metrizable spaces, and let $f$ be an approximable mapping (IX, p. 115, exerc. 22) of $X$ into $Y$. Now prove that for every Borel set $B \subset Y, f^{-1}(B)$ is approximable in $X$ (cf. IX, p. 112, exerc. 6). Deduce that if $g$ is a Borel mapping of $Y$ into a metrizable space $Z$, $g \circ f$ is approximable. Give an example of a continuous mapping $f$ and an approximable mapping $g$ such that $g \circ f$ is not approximable (cf. IX, p. 122, exerc. 14 b)).

Appendix I

1) Let $X$ be an uncountable discrete space, and let $X'$ be its Alexandroff compactification. Show that $X'$ is a Lindelöf space in which there are open subspaces which are not Lindelöf spaces.

App. 2

EXERCISES

TG IX.125

2) a) Show that the product of a Lindelöf space and a compact space is a Lindelöf space (cf. I, p. 70, prop. 17).

b) Give an example of a Lindelöf space X all of whose subspaces are paracompact and such that $X \times X$ is not a Lindelöf space (cf. IX, p. 113, exerc. 15).

3) Let X be the space obtained by endowing $\mathbf{R}^2$ with the topology generated by the open sets of the usual topology of $\mathbf{R}^2$ and the complement in $\mathbf{R}^2$ of the open half-line consisting of the points $(0, y)$ such that $y > 0$ (cf. I, p. 103, exerc. 20 c)). Show that X is a Lusin space which is not regular (to see that X is Lusin, consider the Polish space which is the sum of the closed half-plane $y \leqslant 0$ and the open half-plane $y > 0$ in $\mathbf{R}^2$).

4) Let X be a regular space, A a subset of X which is a Lindelöf subspace, and which moreover has the following properties: the subspace $X - A$ is paracompact, and every neighbourhood of A in X contains a neighbourhood of A which is closed in X. Show that X is then paracompact. (Let $\mathscr{R}$ be an open covering of X, $\Sigma$ a countable subset of $\mathscr{R}$ forming a covering of A, U the union of the sets of V, W an open neighbourhood of A such that $\overline{W} \subset U$. Use the fact that $X - W$ is paracompact, as well as exerc. 25 of IX, p. 107 and 29 a) of IX, p. 108.)

5) a) There exists in the interval $I = [0, 1]$ of $\mathbf{R}$ an uncountable subset Y such that every compact subset of I contained in Y is countable (IX, p. 114, exerc. 17 d)). Let $X_0$ be the closure of Y in I, and X the topological space obtained by taking as open sets in X the sets $U \cup S$, where is open for the topology induced on $X_0$ by that of I, and S is an arbitrary subset of Y. Show that X is regular, and that there is a basis for the topology of X such that every point of X belongs to only countably infinitely many sets of this basis.

b) Show that every open set V of X which contains $M = X - Y$ has a countable complement in X. Deduce that X is not metrizable (M is not a countable intersection of open sets), and that X is a Lindelöf space.

c) Show that every subspace Z of X is paracompact (but not necessarily a Lindelöf space). (Apply exerc. 4 by taking $A = Z \cap M$.)

d) Let $Y_0$ be the set Y endowed with the topology induced by that of I. Show that the product space $X \times Y_0$ is not normal though $Y_0$ is metrizable. (Let $\Delta$ be the closed set in $X \times Y_0$ formed by the points $(x, x)$, where $x \in Y$; $\Delta$ and the closed set $M \times Y_0$ have no point in common. Show that every open neighbourhood U of $\Delta$ in $X \times Y_0$ necessarily meets every open neighbourhood of $M \times Y_0$. To this end, let $W_n$ be the set of $x \in Y$ such that the set of $(x, y)$ with $y \in Y, |x - y| < 1/2n$, belongs to V; the $W_n$ cover Y. Using b), show that there is an index k such that $\overline{W}_k$ meets M; let $x \in \overline{W}_k \cap M, y \in Y$ such that $|x - y| < 1/2k$; show that every neighbourhood of $(x, y)$ in $X \times Y_0$ meets V).