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.
Uncountable limit point of uncountable Set (Munkres Topology)
4
$\begingroup$
general-topology
algebraic-topology
1 Answers
5
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 $C=\bigcup_{B\in\mathscr{B}_0}(A\cap B)\;.$
- Show that $C$ is countable.
- 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 $A$.
-
0@beginner: You’re welcome; I’m glad to hear it. – 2012-11-10