Given any uncountable subset $S$ of $\mathbb{R}$ , does there exist a non empty perfect set $A$ such that $A \subseteq S$ ?

  • $\begingroup$ What's a perfect set? $\endgroup$ – Henrik Jun 3 '17 at 8:48
  • $\begingroup$ I think that $S =\mathbb{R}\setminus \mathbb{Q}$ does not contain any nonempty perfect set. $\endgroup$ – Rigel Jun 3 '17 at 8:50
  • $\begingroup$ It has...problem in Rudin ...chapter 2 qstn 18 $\endgroup$ – CoffeeCCD Jun 3 '17 at 8:52
  • $\begingroup$ math.stackexchange.com/questions/1064/… $\endgroup$ – CoffeeCCD Jun 3 '17 at 9:13
  • $\begingroup$ Oh yes, sorry, you are right! $\endgroup$ – Rigel Jun 3 '17 at 9:15

It depends on the use of the axiom of choice AC.

wiki: Perfect set property

a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150).

The axiom of choice implies the existence of sets of reals that do not have the perfect set property.

This says that in a polish space (like the real numbers) AC implies, there are sets without the perfect set property, which means they are not countable and do not contain a perfect subset.

However, in Solovay's model, which satisfies all axioms of ZF but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary.


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