Exercise 12, § 4

Let $X$ be a Hausdorff topological space. For every pair $(K, U)$ formed by a compact set $K$ and an open set $U$ in $X$, let $I(K, U)$ be the set of subsets $M \subset X$ such that $K \subset M \subset U$, and let $\mathscr{T}$ be the topology on $\mathfrak{P}(X)$ generated by the set of subsets $I(K, U)$ of $\mathfrak{P}(X)$.

a) Show that each of the sets $I(K, U)$ is both open and closed in $\mathfrak{P}(X)$; from this, deduce that $\mathfrak{P}(X)$, equipped with the topology $\mathscr{T}$, is a completely regular and totally disconnected space.

b) For the mapping $M \mapsto \mathbf{C}M$ of $\mathfrak{P}(X)$ onto itself to be continuous (for the topology $\mathscr{T}$), it is necessary and sufficient that $X$ be compact.

c) Take $X$ to be the interval $[0,1]$ of $\mathbf{R}$. Show that the mappings $(M,N) \mapsto M \cup N$ and $(M,N) \mapsto M \cap N$ of $\mathfrak{P}(X) \times \mathfrak{P}(X)$ into $\mathfrak{P}(X)$ are not continuous for the topology $\mathscr{T}$.

d) Assume $X$ to be locally compact. Show that the topology induced by $\mathscr{T}$ on the set of compact subsets of $X$ is finer than the topology deduced from a uniform structure on $X$ by the procedure of Exer. 5 of GT, II, §1; these two topologies cannot be identical unless $X$ is a discrete space.