Logical meaning of a sentence

Question

There is one question, which says, "Give an example of a separable space $\left( X, \mathscr{T} \right)$ in which there is an uncountable set which does not contain any of its limit point".

Now, the last part, which says, "A set which does not contain any of its limit points" is a bit confusing. Is it equivalent to "No point in the set is a limit point of the set"?

The confusion arises because, maybe the set has no limit points at all (and hence no point of the set is a limit point) and then vacously, it contains all of its limit points (which is exact opposite of the statement).

Answer 1

If a set has no limit points, then it simultaneously contains all its limit points and none of its limit points. There is no contradiction here, and that's exactly the kind of example I would look for. At least at first. Uncountable is tricky, though, so you may instead have to look for an example which has limit points but doesn't contain them.

Answer 2

You are being asked to give an example of a separable space $X$ which has an uncountable subset $Y$ such that any limit point $y$ of $Y$ in $X$ has $y \not \in Y$. If $Y$ somehow had no limit points, you'd be done.