[yorwba]

That's a very nice intuition, but for the wrong concept. What you have been describing is the difference between a dense set (almost no holes) and a nowhere-dense set (holes everywhere).

It turns out that there is a nowhere-dense set that is still uncountably infinite: https://en.wikipedia.org/wiki/Cantor_set

[waqf]

No, iamlucaswolf is correct, describing a countable dense set (the dyadic rational points) and an uncountable dense set.