The Growth of the Subuniform Ultrafilters on ω_1
Extending real-valued functions in βκ
An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.
Remote points in spaces with π-weight
The regular open algebra of βRR is not equal to the completion of P(ω)/fin
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.
Two results on special points
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.
Representing free Boolean algebras
Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra...
Are initially -compact separable regular spaces compact?
We investigate the question of the title. While it is immediate that CH yields a positive answer we discover that the situation under the negation of CH holds some surprises.
The measure algebra does not always embed
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.
More on tie-points and homeomorphism in ℕ*
A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique fixed point...
Is 𝓟(ω) a subalgebra?
We consider the question of whether 𝒫(ω) is a subalgebra whenever it is a quotient of a Boolean algebra by a countably generated ideal. This question was raised privately by Murray Bell. We obtain two partial answers under the open coloring axiom. Topologically our first result is that if a zero-dimensional compact space has a zero-set mapping onto βℕ, then it has a regular closed zero-set mapping onto βℕ. The second result is that if the compact space has density at most ω₁, then it will map onto...
More about spaces with a small diagonal
Hušek defines a space X to have a small diagonal if each uncountable subset of X² disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω₁ which has a small diagonal will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable; in particular, Juhász and Szentmiklóssy proved...
Two-to-one continuous images of ℕ*
A function is two-to-one if every point in the image has exactly two inverse points. We show that every two-to-one continuous image of ℕ* is homeomorphic to ℕ* when the continuum hypothesis is assumed. We also prove that there is no irreducible two-to-one continuous function whose domain is ℕ* under the same assumption.
Ordinal remainders of classical ψ-spaces
Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain of infinite subsets of ω, there exists , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain , hence a ψ-space with Stone-Čech remainder...
Reflecting Lindelöf and converging ω₁-sequences
We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ω-sequence or a non-trivial converging ω₁-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ω₁-sequences is first-countable and, in addition,...
Property D and pseudonormality in first countable spaces
In answer to a question of M. Reed, E. van Douwen and M. Wage [vDW79] constructed an example of a Moore space which had property D but was not pseudonormal. Their construction used the Martin’s Axiom type principle . We show that there is no such space in the usual Cohen model of the failure of CH.
A new Lindelöf space with points
We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality which has points . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.
On van Douwen spaces and retracts of
Eric van Douwen produced in 1993 a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of . We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of expanding on earlier work of Balcar and Błaszczyk (1990) and Simon (1987).
Asymmetric tie-points and almost clopen subsets of
A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of . One especially important application, due to Veličković, was to the existence of nontrivial involutions on . A tie-point of has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set...
Skula spaces
Čech-Stone remainders of spaces that look like
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