Definition:Isolated Point (Metric Space)

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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 $S$ if and only if there exists an open $\epsilon$-ball of $x$ in $M$ containing no points of $S$ other than $a$:

$\exists \epsilon \in \R_{>0}: B_\epsilon \left({a}\right) \cap S = \left\{{a}\right\}$

That is:

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


Isolated Point in Space

When $S = A$ this reduces to:

$a \in A$ is an isolated point of $M$ if and only if there exists an open $\epsilon$-ball of $x$ containing no points other than $a$:

$\exists \epsilon \in \R_{>0}: B_\epsilon \left({a}\right) = \left\{{a}\right\}$

That is:

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


Metric Space as a Topological Space

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

$\tau{\left({A, d}\right)} := \left\{{B_\epsilon \left({a}\right): \epsilon \in \R_{>0}, a \in A, B_\epsilon \left({a}\right) \subseteq S}\right\}$

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.