Does every uncountable set of reals contain an interval?

[dreamtheater]

Let S be an uncountable subset of the reals.

Then does S always contain at least one interval (whether it be open/closed/half-open/rays/etc..)?

Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?

My intuition is that, if any subset of the reals contains no intervals, then it must be denumerable, but this might be wrong.

 

[sylas]

Let S be an uncountable subset of the reals.

Then does S always contain at least one interval (whether it be open/closed/half-open/rays/etc..)?

Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?

My intuition is that, if any subset of the reals contains no intervals, then it must be denumerable, but this might be wrong.

Consider the set of all irrational numbers.

 

[lurflurf]

False!
The Cantor, the irrationals, and the transendential numbers are obvious counter examples.

 

[Preno]

Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?

Take any interval. It has length d. But for sufficiently high n, 3^{-n} < d. Therefore, by construction, the Cantor set contains no intervals of length d.