§1 is concerned with variations on the theme of an ordinal compactification of the integers. Several applications are found, yielding, for instance, an example previously known only modulo the continuum hypothesis, and a counterexample to a published assertion.§2 is concerned with zero-one sequences and §3 with spaces built from sequential fans. Of two old problems of Cech, one is solved and one partly solved.Since the sections are more or less independent, each will have its own introduction. Sequential spaces form the connecting thread, although not all the examples are concerned with them.
Spaces such as ßN and /*i provide ready examples of separable compactHausdorff spaces which are not sequential^). But these are of "large" cardinality, i.e. 2C. The space a>x +1 with the order topology is a nonsequential, compact Hausdorff space of "small" cardinality, i.e. K,, but, unfortunately, it is not separable. This leads one naturally to ask if there is a nonsequential, but separable, compact Hausdorff space of small cardinality. Such a space can be produced simply by conjoining known theorems as follows.Magill [M, Theorem 2.1] showed that if any Hausdorff space Kis the continuous image of ßX\X, with A'locally compact Hausdorff, then there is a compactification yX of X with yX\X homeomorphic to K. Parovicenko [P, Theorem 1] proved that every compact Hausdorff space of weight á Xi is the continuous image of ßN\N. From these results one obtains Example 1.1. There is a compactification yN of N with yN\N homeomorphic to coy +1, and hence there is a nonsequential, but separable, compact Hausdorff space of cardinality Xj.By providing a specific construction of the space yN, which is done below, we can assure (modulo the continuum hypothesis (CH)) that no sequence in TV converges to cuj g yN. Then by modifying the topology of yN at the point oeu we get Example 1.2. (CH) There is a sequentially compact, Hausdorff c-space (2) which