Or, four questions that I think are equivalent, but I don't know if will be thought to make sense:
- Are there an odd number of numbers in $[0, 1]$?
- Are there an even number of numbers in $[0, 1)$?
- Are there an even number of numbers in $(0, 1]$?
- Are there an odd number of numbers in $(0, 1)$?
My basic reasoning is that there must be the same number of (real) numbers in $[0, 0.5)$ as in $(0.5, 1]$, meaning that there are an even number of numbers in their union, and therefore additionally including $0.5$ there are an odd number of numbers in $[0, 1]$.
Seems straightforward, but uncountable sets aren't, and therefore I don't know if their parity is a sensible concept to talk about...
The concepts of odd and even only apply (directly) to finite numbers. There isn't a direct way to talk about infinite (countable or uncountable) sets having an even or odd number of elements.
If you want to talk about even and odd in the infinite setting, you can look into even and odd ordinals. The ordinals are a different way to generalize the natural numbers into numbers that are "infinite."
To get you started, $\omega$ is the size of the set of $\mathbb{N}$ (the set of natural numbers). This is the smallest infinite ordinal and its even. Then, you can take $\omega + 1$, $\omega + 2$, etc... which are odd, even, etc... However, from the perspective of "size", all of these ordinals (and a lot more) are all $\aleph_0$ in "size". There are ordinals of bigger sizes, but you need to do a fair amount of work to get to them.
Ordinals are extremely cool, I highly recommend looking into them.