I remember the characterization of open subsets of $\mathbb{R}$ as a countable union of disjoint open intervals. I was thinking about whether this allows us to characterize closed subsets as a countable union of disjoint closed intervals. As we know, closed subsets are just complements of open subsets.
I had an idea of starting with an open interval at $(a,b)$, then looking at the next open interval to the right at $(c,d)$, and concluding that $[b,c]$ belongs to the closed subset. But this idea can break down, for example, if the open subset contains the intervals $(-1,0), (1/2,1), (1/8,1/4), (1/32,1/16), \ldots$. In this case, there is no next open interval to the right of $(-1,0)$. How can we modify this idea to characterize closed subsets?
Best Answer
It’s not true that all closed sets are countable unions of disjoint closed intervals: the middle-thirds Cantor set is a counterexample. It is closed and nowhere dense in $\Bbb R$, so it contains no non-trivial closed interval. That is, the only closed intervals that it contains are degenerate ones of the form $[x,x]=\{x\}$. And since it is uncountable, it isn’t a countable union even of these degenerate closed intervals.