Finite Subset of Metric Space has no Limit Points
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $X \subseteq A$ such that $X$ is finite.
Then $X$ has no limit points.
Proof
Let $x \in X$.
From Point in Finite Metric Space is Isolated, $x$ is an isolated point.
The result follows by definition of isolated point:
- $x$ is an isolated point if and only if $x$ is not a limit point.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 22$