Eric van Douwen [vD93] produced a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of βN. We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of βN expanding on earlier of work in [BBS94, Sim87].

It is well known that a sum (coproduct) of a family {Xi : i ∈ I} of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully u...

It is shown in this paper that if BG is the Stone-ech compactification of a group G, and G satisfying a certain condition, then there is a weakly recurrent point in BG which is not almost periodic, and if another condition will be added, then there is a recurrent point in 8G which is not almost periodic point.

Let β(N) denote the Stone–Čech compactification of the set N of natural numbers (with the discrete topology), and let N∗ denote the remainder β(N)−N. We show that, interpreting modal diamond as the closure in a topological space, the modal logic of N∗ is S4 and that the modal logic of β(N) is S4.1.2.

We prove that the number of near-coherence classes of non-principal ultrafilters on the natural numbers is either finite or 2c. Moreover, in the latter case the Stone-Čech compactification βω of ω contains a closed subset C consisting of 2c pairwise non-nearly-coherent ultrafilters. We obtain some additional information about such closed sets under certain assumptions involving the cardinal cha...

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usu...