Accumulation point

With the punctured neighbourhood we can say precisely when a point is “surrounded” by points of a set.

Definition — Accumulation point

Let $D\subseteq\mathbb{R}$. The point $x_0\in\mathbb{R}$ is an accumulation point of $D$ if in every punctured neighbourhood of $x_0$ there falls at least one point of $D$: $\forall\,\delta>0:\quad \dot I_\delta(x_0)\cap D \ne \emptyset.$ Equivalently: every neighbourhood of $x_0$ contains infinitely many points of $D$.

The two formulations are equivalent: if in every punctured neighbourhood, however small, there always falls at least one point of $D$, then infinitely many points necessarily fall there (as close as we like to $x_0$).

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Topics: Limits
Concepts: Neighbourhood · Accumulation point
Methods: Accumulation point
Skills: Proving