Exercise 26, § 11

Let $X$ be a topological space. a) We equip the set $\mathfrak{P}_0(X)$ with one of the topologies $\mathfrak{T}_\Omega, \mathfrak{T}_\varnothing$ (I, p. 91, exer. 7) or $\mathfrak{T}_\Theta$ (I, p. 101, exer. 12). Now prove that if a subset $\mathfrak{B}$ of $\mathfrak{P}_0(X)$ is connected and if every set $M \in \mathfrak{B}$ is connected, then the union $\bigcup_{M \in \mathfrak{B}} M$ is connected. b) Let $\mathfrak{S}$ be a subset of $\mathfrak{P}_0(X)$ containing the set of all finite non-empty subsets of $X$. Now prove that if $X$ is connected, the same is true of $\mathfrak{S}$ (use exer. 7 b) of I, p. 91 and prop. 8 of I, p. 83, by considering the mappings $(x_1, x_2, \ldots, x_n) \mapsto \{x_1, x_2, \ldots, x_n\}$.