The cardinality of countably compact Hausdorff spaces (Q1097537)

A space is crowded if it has no isolated points. It is well-known that a countably compact regular space which is not sequentially compact or which is crowded has cardinality at least \({\mathfrak c}(=2^{\omega})\). The authors prove that \(\omega_ 1=\min \{| X|:\) X is a crowded sequentially compact Hausdorff space\(\}\). Let \(\mu_{cc}\) be the least cardinality of a crowded countably compact Hausdorff separable space, \(\mu_{sc}\) the least cardinality of a crowded sequentially compact Hausdorff separable space, and \(\mu_{ns}\) the least cardinality of a Hausdorff countably compact space that is not sequentially compact. In addition, the authors prove the interesting fact that for any of these \(\mu\) 's each of \(\omega_ 1=\mu <{\mathfrak c}\), \(\omega_ 1<\mu <{\mathfrak c}\), and \(\omega_ 1<\mu ={\mathfrak c}\) is consistent.