Cardinality of Real numbers in arbitrary small interval.

Question

I have been learning about the cardinality of Real numbers and it got me thinking. Since the cardinality of real numbers is uncountable and is the same as the cardinality of real numbers in the interval $(0,1)$ or for that matter any given interval.

So my question is how small this interval can get and still retain the same cardinality. Does $[0, \epsilon)$ have some cardinality as real numbers as epsilon gets smaller and smaller. It is trivial to see that in the limiting case, the set contains only one element.

So the question is how small can epsilon get before the cardinality of the set $[0, \epsilon)$ gets smaller than that of Real numbers.

Answer 1

So the question is how small can epsilon get before
the cardinality of the set $[0, \epsilon)$ gets smaller than that of real numbers.

If $\epsilon \gt 0$, it can never get smaller than the cardinality of the real numbers.

That means if $\epsilon \gt 0$ the interval $[0,\epsilon)$ has the same cardinality as $[0,1)$, or as the real numbers.