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As we know,

For every $T_2$, first countable, compact space, its cardinality is not more than $2^\omega$. (See chapter 3 of Engelking's book.)

However, I want to know whether the result is same for the $T_2$, first countable, countably compact space, i.e., for every $T_2$, first countable, countably compact space, its cardinality is also not more than $2^\omega$?

Thanks for any help:)

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    $B$TW tagging a question both [tag:elementary-set-theory] and [tag:set-theory] seems strange to me.2012-07-06

1 Answers 1

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For any uncountable regular cardinal $\kappa$ consider the subspace $S=\{\alpha\in \kappa: cof(\alpha)=\omega\}$ with the order topology. $S$ has cardinality $\kappa$ and it is easy to see that satisfies all your requirements.

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    @John: All the ordinals below $\kappa$ whose cofinality is $\omega$ (if $\kappa=\omega_2$ then this means all countable ordinals, and ordinals such as $\omega_1+\omega,\omega\cdot\omega_1$ and so on).2012-07-06