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.