A copy of the Cantor set is a space homeomorphic to $2^{\omega}$.


Suppose that $X$ is a Hausdorff space that contains a copy $C^{\prime}$ of the Cantor set. Let $U$ be a nonempty subset open in $C^{\prime}$, also let $D$ be a countable set dense in $X$ such that $D\cap U$ is dense in $U$. Does anyone have any idea how to prove that the set of accumulation points of $D\cap U$ is infinite?



Subsets of the Cantor set
To answer the question: every point of $U$ is an accumulation point of $D\cap U$, hence there are continuum many accumulation points. The ambient space $X$ plays no role here; everything takes place in the Cantor set.


