Exercise 13, § 2

a) On the set $\mathbf{R}$ of real numbers, consider the topology $\mathscr{T}$ defined as follows: for each $y > 0$, $U_y(x)$ denotes the union of the intervals $[x, x + y[$ and $] - x - y, - x[$; $\mathscr{T}$ is the topology for which the $U_y(x)$ form a fundamental system of neighbourhoods of $x$ as $y$ runs through the set of real numbers $> 0$. Let $X$ be the space obtained by endowing the interval $[-1, +1]$ with the topology induced by $\mathscr{T}$. Show that $X$ is compact. (Consider a filter $\mathfrak{F}$ on $X$; if $x \in X$ is a cluster point of $\mathfrak{F}$ with respect to the topology of the real line, show that either $x$ or $-x$ is a cluster point of $\mathfrak{F}$ with respect to $\mathscr{T}$.)

b) Every point of $X$ has a countable fundamental system of neighbourhoods, and $X$ has a countable dense subset, but $X$ is not metrizable (show that its topology has no countable base).

c) Let $A$ be an open subset of $X$. Show that $A$ is the union of a countable family $(I_n)$ of open intervals contained in $[0, 1]$, the intervals $-I_n$, a subset of the set of left-hand points of the intervals $I_n$ and $-I_n$, and possibly the point $+1$. Hence show that $A$ is the union of a countable family of closed subsets of $X$.

d) Let $Y$ be the set $I \times \{1, 2\}$, where $I$ is the interval $[0, 1]$ of $\mathbf{R}$; let $X_i$ denote $I \times \{i\}$ ($i = 1, 2$) and let $f$ denote the bijection $(x, 1) \to (x, 2)$ of $X_1$ onto $X_2$. For each $x \in I$, let $\mathscr{B}((x, 2))$ denote the set of subsets consisting only of $\{(x, 2)\}$, and let $\mathscr{B}((x, 1))$ denote the set of subsets of the form $V_1 \cup (f(V_1) - \{x, 2\})$, where $V_1 = V \times \{1\}$ and $V$ runs through a fundamental system of neighbourhoods of $x$ in $I$. Show that for each $y \in Y$, $\mathscr{B}(y)$ is a fundamental system of neighbourhoods of $y$ for a topology on $Y$. Endowed with this topology, $Y$ is compact and every point of $Y$ has a countable fundamental system of neighbourhoods, but $Y$ has no countable dense subset. Every closed subset of $Y$ contained in $X_2$ is finite; hence show that the compact subset $X_1$ of $Y$ has no countable fundamental system of neighbourhoods. If $R$ is the equivalence relation on $Y$ whose classes are the set $X_1$ and the points of $Y - X_1$, $R$ is closed and every equivalence class mod $R$ is compact, but there is a point in $Y/R$ which has no countable fundamental system of neighbourhoods [cf. § 4, Exercise 24 a)].