Accumulation Point

tags:definition |topology

Accumulation Point

Context

Statements

Definition (@Eck23, Definition 1.5, @Men90, Defintion 6.6). Let $(X,d)$ be a metric space . A point $s\in S$ is said to be an accumulation point of $S$ if, for every $\epsilon >0$, the intersection of $S$ with the open ball $B_{<\epsilon }(s)$ is non-empty, i.e., $S\cap B_{<\epsilon }(s)\setminus \{s\})\ne \varnothing$.

Remarks

The definition can also be restated in term of metric neighborhood: $s$ is an accumulation point if every neighborhood of $s$ contains a point of $S$ that is different from $s$.

References