Exercise 7, § 2

If $X$ is any set, let $\mathfrak{P}_0(X)$ denote the set of non-empty subsets of $X$.

a) Let $X$ be a non-empty topological space. Let $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) denote the coarsest topology on $\mathfrak{P}_0(X)$ such that for every non-empty open (resp. closed) subset $A$ of $X$, $\mathfrak{P}_0(A)$ is open (resp. closed) in $\mathfrak{P}_0(X)$. Show that in general these two topologies are not comparable, and that the mapping $x \to \{x\}$ is a homeomorphism of $X$ onto a subspace of $\mathfrak{P}_0(X)$ if this latter set carries either of the topologies $\mathfrak{T}_\Omega, \mathfrak{T}_\Phi$.

b) Let $D$ be a dense subset of $X$. Show that the set $\mathfrak{G}(D)$ of finite non-empty subsets of $D$ is dense in $\mathfrak{P}_0(X)$ for either of the topologies $\mathfrak{T}_\Omega, \mathfrak{T}_\Phi$.

c) If $\mathfrak{P}_0(X)$ is ordered by inclusion, show that the topology $\mathfrak{T}_\Omega$ is coarser than the left topology on $\mathfrak{P}_0(X)$ (§ 1, Exercise 2); that $A \in \mathfrak{P}_0(X)$ is isolated in the topology $\mathfrak{T}_\Omega$ if and only if $A = \{x\}$, where $x$ is an isolated point of $X$; and that $A \in \mathfrak{P}_0(X)$ is isolated in the topology $\mathfrak{T}_\Phi$ if and only if $X$ is a finite discrete space and $A = X$.

d) Prove that if $\mathfrak{P}_0(X)$ carries the topology $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) and

carries the topology which is the product of $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) with itself, then the mapping $(M, N) \to M \cup N$ of $\mathfrak{P}_0(X) \times \mathfrak{P}_0(X)$ into $\mathfrak{P}_0(X)$ is continuous.

e) Show that if $\mathfrak{P}_0(X)$ carries the topology $\mathfrak{T}_\Phi$, then the mapping $M \to \overline{M}$ is continuous.

f) Let $X, Y$ be two topological spaces, $f : X \to Y$ a continuous map. Show that if both $\mathfrak{P}_0(X)$ and $\mathfrak{P}_0(Y)$ carry the topology $\mathfrak{T}_\Omega$, and if both carry the topology $\mathfrak{T}_\Phi$, then the mapping $M \to f(M)$ of $\mathfrak{P}_0(X)$ into $\mathfrak{P}_0(Y)$ is continuous.

g) Let $X, Z$ be two topological spaces. Show that a mapping $f : Z \to \mathfrak{P}_0(X)$ is continuous when $\mathfrak{P}_0(X)$ carries the topology $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) if and only if the set of all $z \in Z$ such that $f(z) \cap A \neq \emptyset$ is closed (resp. open) in $Z$ for each closed (resp. open) subset $A$ of $X$.