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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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BTW tagging a question both elementary-set-theory and set-theory seems strange to me. – Martin Sleziak Jul 6 '12 at 5:11
up vote 6 down vote accepted

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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I'm not familar with such space; please explain more:) – Paul Jul 6 '12 at 3:06
@John This is a slight modification of Example 3.10.16 of Engelkin book. – azarel Jul 6 '12 at 3:15
@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). – Asaf Karagila Jul 6 '12 at 5:15

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