Infinite Sequence in Countably Compact Space has Accumulation Point

Corollary to Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point

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

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $S$.

Then $\sequence {x_n}$ has an accumulation point in $T$.

Proof

Let $A \subseteq S$ be the range of $\sequence {x_n}$:

$A = \set {x_n: n \in \N}$

If $A$ is finite, then consider the equality:

$\ds \N = \bigcup_{y \mathop \in A} \set {n \in \N: x_n = y}$

Therefore, there exists a $y \in A$ such that $\set {n \in \N: x_n = y}$ is an infinite set.

Hence, $y$ is an accumulation point of $\sequence {x_n}$.

Otherwise, $A$ is countably infinite.

Then $A$ has an $\omega$-accumulation point in $T$.

It follows that $\sequence {x_n}$ has an accumulation point in $T$.

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$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties