Continuum Hypothesis for closed sets

Question

In A Beginner's Guide to Modern Set Theory [page 48], the author says:

[Cantor] did prove that every closed uncountable subset of $\mathbb R$ has cardinality $2^{\aleph_0}$...

... but I cannot find the proof anywhere, and it doesn't seem trivial. Can someone help me, either by pointing me to a proof, or proving it directly?

The argument is as outlined in the answer. For details, see here. — Andrés E. Caicedo, Nov 29 '14 at 08:01
More generally, any uncountable Borelian subset of a Polish space has cardinality at least $2^{\aleph_0}$. See here: https://chiasme.wordpress.com/2013/08/05/the-continuum-hypothesis-for-borelian-sets/ — Seirios, Nov 29 '14 at 08:39

score 3 · Accepted Answer · edited Nov 29 '14 at 07:58

3

A perfect set is a set which is equal to the set of its limit points. (In particular, it's closed.)

Any nonempty perfect subset of $\mathbb{R}$ has cardinality $2^{\aleph_0}$.

The Cantor Bendixson Theorem states that any closed subset of $\mathbb{R}$ can be written (uniquely) as the disjoint union of a countable set and a perfect set. Hence an uncountable closed set contains a nonempty perfect set, and hence has cardinality $2^{\aleph_0}$.

(The above two statements about $\mathbb{R}$ work for Polish spaces more generally.)

edited Nov 29 '14 at 07:58

Andrés E. Caicedo

79,201

answered Nov 29 '14 at 07:16

aes

7,653

The definition of limit point excludes isolated points. – aes Nov 29 '14 at 09:41

Continuum Hypothesis for closed sets

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