If I have an uncountable subset $A \in \mathbb{R}$, and we assume A is nonempty, does it follow that every point within $A$ is a limit point of $A$ from the density of $\mathbb{Q}$ in $\mathbb{R}$ (i.e. between any two real numbers there exists one rational number, and by extension infinitely many rationals)?
As far as I can tell, this logic would not hold for all countable subsets, since I believe, for example, that $\mathbb{Z}$ has no limit points whatsoever.
I'm trying to straighten out the concept of limit points in my head, and I'd love some clarification of my logic. Many thanks!