Let $I$ be an uncountable set of indices and $U_\alpha\subset\mathbb{R}^3$ be an open subset for each $\alpha\in I$. We know that $U=\bigcup_{\alpha\in I} U_\alpha$ is an open subset.
My question is: does there exist a countable subset $J \subset I$ such that $U=\bigcup_{\alpha\in J} U_\alpha$?
In general it is not true without the openness condition.
Thanks!
As suggested by Chandru, this is a `named' property: Lindelöf space.
Every $\sigma$--compact set is Lindelöf: Let $K_0\subset\cdots K_n\to\subset U$ be an increasing sequence of compact subsets with $\bigcup_{n\ge0}K_n=U$. Each $K_n$ has a finite open cover. So $U$ has a countable open cover.