Let {Ωα : α ∈ I} be an arbitrary family of closed sets Ωα ⊆R^d with an index set I.
(a) Prove that ⋂Ωα is a closed set. [7]
(b) Set d = 2 and show by construction of a counterexample that ⋃ Ωα is not necessarily closed. [2]
I understand the definitions of closed sets (contains all of its limit points) and that if a set is compact then it is closed but cant seem to work this out.
(I). Let $F$ be a family of closed subsets of $\Bbb R^n.$
Let $\sigma=(p_n)_{n\in \Bbb N}$ be any convergent sequence of members of $\cap F,$ converging to $p.$ Then for each $f\in F,$ all the members of $\sigma$ belong to $f,$ and $f$ is closed, so $p\in F.$ Since $p$ belongs to every member of $F,$ therefore $p$ belongs to $\cap F.$
(II). If $U$ is any non-closed subset of $\Bbb R^n,$ consider that $F=\{\{p\}:p\in U\}$ is a family of closed sets of $\Bbb R^n$ but $\cup F =U$ is not closed.