Isolated point and membership of the domain

Being an accumulation point and belonging to the set are two independent properties: they must be kept clearly distinct.

Remark — Difference from "isolated point" and from "belonging to $D$"

• An isolated point of $D$ is a point of $D$ that has a neighbourhood in which no other point of $D$ falls. An isolated point is not an accumulation point.
• An accumulation point may not belong to $D$: for example $0$ is an accumulation point of $D = (0,1]$ even though it is not inside $D$.
• A point $x_0\in D$ may or may not be an accumulation point: in $D = \{1\}\cup [2,3]$, the point $1$ belongs to $D$ but is isolated, whereas $2$ belongs to $D$ and is an accumulation point.

In short: “accumulation point” describes how the points of $D$ cluster around $x_0$, and it is independent of whether or not $x_0$ is an element of $D$.

Links

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