391: Topological Space Is Countably Compact if It Is Sequentially Compact

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A description/proof of that topological space is countably compact if it is sequentially compact

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of countably compact topological space.
• The reader knows a definition of sequentially compact topological space.
• The reader admits the proposition that any topological space is countably compact if and only if each infinite subset has an $\omega$-accumulation point.

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Target Context

• The reader will have a description and a proof of the proposition that any topological space is countably compact if the space is sequentially compact.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

Any topological space, $T$, is countably compact if $T$ is sequentially compact.

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2: Proof

Let suppose that $T$ is sequentially compact. We will prove that each infinite subset, $S\subseteq T$, has an $\omega$-accumulation point, $p\in T$, which will imply that $T$ is countably compact, by the proposition that any topological space is countably compact if and only if each infinite subset has an $\omega$-accumulation point.

Let any infinite subset be $S\subset T$. Let us choose any sequence of points from $S$, $p_1,p_2,...$, which is possible because $S$ is infinite. There is a subsequence, $p_1^{\prime },p_2^{\prime },...$, that converges to a point, $p\in T$. $p$ is an $\omega$-accumulation point of $S$, because for any neighborhood, $U_p\subseteq T$, there is an $i_0$ such that $p_i^{\prime }\in U_p$ for each $i$ such that $i_0<i$, which means that $U_p\cap S$ is infinite.

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3: Note

The converse is not true in general. The converse with the requirement that $T$ is 1st-countable is true, as is proved in another article.

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References

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