On $T_2$, first countable, countably compact space

Question

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:)

Answer 1

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.