Examples of accumulation points

Example — Typical sets

• $\mathbb{N} = \{0,1,2,\ldots\}\subset\mathbb{R}$: all its points are isolated; no finite real is an accumulation point (only $+\infty$, if we allow points at infinity).
• $\mathbb{Z}\subset\mathbb{R}$: as above, all points are isolated.
• $\mathbb{Q}\subset\mathbb{R}$: every real number is an accumulation point of $\mathbb{Q}$ (between two distinct reals there exist infinitely many rationals).
• $D = \{1/n : n\in\mathbb{N},\ n\ge 1\}$: all the points $1/n$ are isolated; $0\notin D$ is an accumulation point.
• $D = [0,1]\cup\{3\}$: every point of $[0,1]$ is an accumulation point; $3$ is isolated.
• $D = (0,1)\cup(1,2)$: every point of $[0,2]$ (endpoints included) is an accumulation point, including $1$ even though it does not belong to $D$.

These examples show the three typical behaviours: sets made only of isolated points (such as $\mathbb{N}$ and $\mathbb{Z}$), “dense” sets in which every real is an accumulation point (such as $\mathbb{Q}$), and sets with accumulation points that lie on the “boundary” but do not belong to the set.

Links

Topics: Limits
Concepts: Number sets · Accumulation point · Isolated point
Skills: Reasoning by cases