We give an example of an extremally disconnected compact Hausdorff space with an open continuous selfmap such that the fixed point set is nonvoid and nowhere dense, respṫhat there is exactly one nonisolated fixed point.
The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss errors in previous results on the embeddability of the measure algebra.
Two compact spaces are co-absoluteif their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω and βℝ are not co-absolute.
We show that there is a nowhere ccc σ-compact space which has a remote point. We show that it is consistent to have a non-compact σ-compact separable space X such that every point of the remainder is a limit of a countable discrete subset of non-isolated points of X. This example shows that one cannot prove in ZFC that every locally compact non-compact space has discrete weak P-points.
We consider the spaces called , constructed on the set of all finite sequences of natural numbers using ultrafilters to define the topology. For such spaces, we discuss continuity, homogeneity, and rigidity. We prove that is homogeneous if and only if all the ultrafilters have the same Rudin-Keisler type. We proved that a space of Louveau, and in certain cases, a space of Sirota, are homeomorphic to (i.e., for all ). It follows that for a Ramsey ultrafilter , is a topological group....