# Real Analysis/Limit Points (Accumulation Points)

## DefinitionEdit

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 ${\displaystyle A}$ .