# Accumulation Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range

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## Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $\sequence {x_n}$ be a sequence of distinct terms of $S$.

Let $\alpha$ be an accumulation point of $\sequence {x_n}$.

Then $\alpha$ is also an $\omega$-accumulation point of $\set {x_n: n \in \N}$.

## Proof

Let $U$ be an open set of $T$ containing $\alpha$.

By definition of accumulation point of $\sequence {x_n}$, $U$ contains infinitely many terms of $\sequence {x_n}$.

As $\sequence {x_n}$ is a sequence of distinct terms: $U$ contains infinitely many elements of $\set {x_n: n \in \N}$.

Thus by definition $\alpha$ is an $\omega$-accumulation point of $\set {x_n: n \in \N}$.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points