[Math] Proof that every closed subset of $\mathbb R$ is finite or countable or continuum.

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I want to prove that every closed subset of $\mathbb R$ is finite or countable or continuum.

I know that for arbitrary subset we can not make similar statements – because of continuum hypothesis.

It looks like I need some machinery to prove mentioned fact. I feel I need somehow characterize all closed subsets of $\mathbb{R}$. For open subsets I know that they are union of disjoint intervals. For closed I do not know anything like that. Intution tells me that some deep unknown for me fact should be used here.

Best Answer

Let $C$ be the set. First, without loss of generality:

  • $C$ is nowhere dense.
  • $C\subseteq [0,1]$

Show that if $C$ is uncountable, there's always an open interval outside of $C$ which divides $C$ into 2 uncountable halves.

If you repeat this process in both halves and so on, compactness implies there's a point in every possible branching, so there's continuum many of them.

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