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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.

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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 $A$.
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Thanks Brian I just do it –  beginner Nov 10 '12 at 4:43
    
@beginner: You’re welcome; I’m glad to hear it. –  Brian M. Scott Nov 10 '12 at 5:01

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