A point $x$ is a cluster point of a set $A\subset \mathbb{R}^p$ if and
only if every neighborhood of $x$ contains infintely many points of
$A$.
I have learned that by definition, a cluster point of a set $S\subseteq X$ is a point $x\in X$ such that for every neighbourhood $U$ of $x$, there is a point $y\in S\cap U$ with $y\ne x$.
Since an open set is a neighborhood of each of its points, no point of an open set can be a limit point of $A$. How can I prove this and what will happen if it were finitely many points?
Best Answer
Suppose there is a neighborhood of $x$ that contains only finitely many points of $A$. If you consider the distances from $x$ to these finitely many points, can you find an open set containing $x$ and no other points of $A$?
For the other direction, I think writing down the negation of "$x$ is a cluster point of A" will be helpful.