# 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 $3.9: 22$