Uncountable limit point of uncountable Set (Munkres Topology)

Question

Let $X$ have a countable basis and $A\subset X$ is uncountable. Would you help me to prove that uncountably many points of A are limit points of A.

Answer 1

HINT: Let $\mathscr{B}$ be a countable base for $X$. Let $\mathscr{B}_0=\{B\in\mathscr{B}:A\cap B\text{ is countable}\}$, and let $$C=\bigcup_{B\in\mathscr{B}_0}(A\cap B)\;.$$

1. Show that $C$ is countable.
2. Show that if $a\in A\setminus C$, then $a$ is a limit point of $A$. (You can prove more here: every open neighborhood of $a$ actually contains uncountably many points of $A$.