# Real Analysis/Limit Points (Accumulation Points)

Let ${\displaystyle (X,d)}$ be a metric space, and let ${\displaystyle A\subset X}$. We call ${\displaystyle x\in X}$ a limit point of ${\displaystyle A}$ if for any ${\displaystyle \epsilon >0}$ there exists some ${\displaystyle y\neq x}$ such that ${\displaystyle y\in B(x,\epsilon )\cap A}$.
We denote the set ${\displaystyle lim(A)}$ the set of all ${\displaystyle x\in X}$ such that ${\displaystyle x}$ is a limit point of ${\displaystyle A}$.