In parallel with the investigation of calibres in Chapter 4, we here examine the extent of classes of compact-calibres of spaces. We determine those classes T of infinite cardinals for which there is a Hausdorff space X(T) such that X(T) does not have compactcalibre a if and only if α∈T; and we show that such a space X(T), when it exists, may be chosen completely regular.

In contrast with its behavior concerning calibre, the cardinal number ω plays no special role with respect to compact-calibre: there are infinite, non-compact, Hausdorff spaces which have compact-calibre ω, and others which do not. Accordingly, the format of Corollary 8.4 is simpler than that of Corollary 4.5 (to which it is similar), and 8.5 and 8.6 are simpler than 4.6 and 4.7, respectively.

The examples we give of spaces with compact-calibre properties prescribed in advance are defined by beginning with the Stone–Čech compactification of a discrete space (which has, like every compact space, compact-calibre α for all α ≥ ω) and discarding from it the (non-principal) ultrafilters with various degrees of non-uniformity. The spaces defined in this way certainly do not have Souslin number equal to ω+, and in fact we do not know whether the spaces X(T) described above can be chosen to satisfy in addition S(X(T)) = ω+.

In Corollary 8.12 we note a curiosity: there are completely regular, Hausdorff, non-compact spaces with compact-calibre α for all α ≥ ω.