Exercise 24, § 11

Let $X$ be a topological space. a) Let the set $\mathfrak{P}_0(X)$ carry one of the topologies $\mathfrak{T}_\Omega, \mathfrak{T}_\Phi$ ($§ 2$, Exercise 7) or $\mathfrak{T}_\Theta$ ($§ 8$, Exercise 12). Show that if a subset $\mathfrak{B}$ of $\mathfrak{P}_0(X)$ is connected and if each $M \in \mathfrak{B}$ is connected, then $\bigcup_{M \in \mathfrak{B}} M$ is connected. b) Let $\mathfrak{S}$ be a subset of $\mathfrak{P}_0(X)$ which contains the set of all finite non-empty subsets of $X$. Show that if $X$ is connected then so is $\mathfrak{S}$ [use Exercise 7 b) of $§ 2$ and Proposition 8 of $§ 4$, considering the mappings $(x_1, \ldots, x_n) \to \{ x_1, \ldots, x_n \}$].