prabau
commented
2 months ago Sure, we can mention this. The wikipedia article https://en.wikipedia.org/wiki/Countably_compact_space (already listed as a reference for the pi-base property) has it as condition (3): "Every sequence in $X$ has a accumulation point in $X$". Feel free to add it.
Note that for sets it's equivalent to use the terminology "accumulation point" or "limit point". But "limit point of a sequence" usually mean a point that is a limit of the sequence. That's why it's better to use the terminology "accumulation point" for a sequence, if that's what is meant (see https://en.wikipedia.org/wiki/Accumulation_point for details).
Maybe it's not necessary, but if you want, we could even be more explicit to avoid any confusion: "Every sequence in $X$ has an accumulation point in $X$ (that is, a point such that each of its neighborhoods contains infinitely many terms of the sequence)."
This is a direct equivalence to "every infinite set in has an $ω$-accumulation point" (at least under axiom of countable choice, or that every infinite set is Dedekind infinite, of course). Does it worth to be mentioned?
Not very sure if this is really necessary, but this criterion using sequences shows convenience to prove that an ordinal $\alpha$ is countably compact if and only if $\text{cf}\ \alpha\neq\omega$. The proof that $\text{cf}\ \alpha\ge\omega_1\Rightarrow\alpha$ is countably compact is to consider the one-point compactification $\alpha+1$.