Definition:Isolated Point (Metric Space)

Definition

Let $M = \left({A, d}\right)$ be a metric space.

Isolated Point in Subset

Let $S \subseteq A$ be a subset of $A$.

$a \in S$ is an isolated point of $S$ iff there exists a neighborhood of $x$ in $M$ containing no points of $S$ other than $a$:

$\exists \epsilon \in \R, \epsilon > 0: N_\epsilon \left({a}\right) \cap S = \left\{{a}\right\}$

That is:

$\exists \epsilon \in \R, \epsilon > 0: \left\{{x \in S: d \left({x, a}\right) < \epsilon}\right\} = \left\{{a}\right\}$

Isolated Point in Subset

When $S = A$ this reduces to:

$a \in A$ is an isolated point of $M$ iff there exists a neighborhood of $x$ containing no points other than $a$:

$\exists \epsilon \in \R, \epsilon > 0: N_\epsilon \left({a}\right) = \left\{{a}\right\}$

That is:

$\exists \epsilon \in \R, \epsilon > 0: \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\} = \left\{{a}\right\}$

Metric Space as a Topological Space

From Metric Induces a Topology we can consider the topology $\vartheta_{\left({A, d}\right)}$ on $A$:

$\vartheta_{\left({A, d}\right)} := \left\{{N_\epsilon \left({a}\right): \epsilon \in \R, a \in A, N_\epsilon \left({a}\right) \subseteq S}\right\}$

and see that the definition given here is compatible with that of the definition for a topological space.

Also see

• Results about isolated points can be found here.