Convergent Sequences in Spaces with Finite Compactness Numbers

摘要

The compactness number of a compact Hausdorff space is the least ordinal k<=ω such that there exists an open subbase satisfying that every cover by elements of the open subbase has a subcover of at most k elements. M.g. Bell and J. van Mill defined spaces X_k with the compactness number k for all k∈ω. In the present paper we show that all nonisolated points in all X_k are the limits of nontrivial sequences. Moreover, we prove that ω ∪{p} has no compactification with the compactness number≤2 for every p∈βωω. But we do not known whether or not it has a compactification with finite compactness number.