Isolated Points

This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

 Definition: A point $c \in A$ is said to be an isolated point of the set $A$ if $c$ is not a cluster point of $A$, i.e., $\exists \delta_0 > 0$ such that $A \cap ( V_{\delta_0} (c) \setminus \{ c \} ) = \emptyset$.
We should note that the definition of an isolated point is the negation of that of a cluster point, and so any point $c \in A$ is either one or the other and cannot be both.
For example, consider the set $(-\infty, -1] \cup \{ 0 \} \cup [1, \infty)$ as illustrated below:
Has an isolated point $c = 0$ in $A$. We can verify this by taking $\delta_0 < 1$. Then any neighbourhood $V_{\delta_0} (c)$ contains only the point $c$, and so $A \cap ( V_{\delta_0} (c) \setminus \{ c \} ) = \emptyset$.