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