A COMPACT F-SPACE NOT CO-ABSOLUTE WITH PN-fV

It will be convenient to call a space X a ParoviZenko space if (cy) X is a zero-dimensional compact space without isolated points, (p) every two disjoint open F,-sets have disjoint closures, and (y) every nonempty GG-set in X has non-empty interior. Compact spaces satisfying (p) are usually called F-spaces, while spaces satisfying (y) are called almost-P spaces. Examples of F-spaces are the extremally disconnected spaces. Examples of almost-P spaces are ncl-sets (and their compactifications). Examples of compact F-almost-P (Parovitenko) spaces are all spaces of the form X* = /3X -X, where X is a locally compact realcompact (respectively, zero-dimensional) space [6,7]. It is well-known that under CH, the continuum hypothesis, all ParovZenko spaces of weight c are homeomorphic [9]. The converse of this result is true, i.e., if all Parovicenko spaces of weight c are homeomorphic, then CH is true [4]. The standard example of a ParoviEenko space of weight c is N*, where N is the discrete space of natural numbers; however, more examples can be produced using spaces of the form (K x RJ)*, where K is a compact zero-dimensional space of weight at most c (e.g. K equal to the Cantor set or N*). The absolute (see [lo] or [16] for surveys) of a regular space X is the unique (up to homeomorphism) extremally disconnected space g(X), which can be mapped