Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|cite|improve this question

1 Answer 1

up vote 5 down vote accepted

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 $A$.
share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.