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1
Content available remote

The regular open algebra of βRR is not equal to the completion of P(ω)/fin

100%
Dow A.
Fundamenta Mathematicae
|
1998
|
tom 157
|
nr 1
33-41
EN
Two compact spaces are co-absolute} if their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω\ω and βℝ\ℝ are not co-absolute.
2
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Extending real-valued functions in βκ

100%
Dow A.
Fundamenta Mathematicae
|
1997
|
tom 152
|
nr 1
21-41
EN
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 $\got c$ and that it is consistent that ω*\{p} is C*-embedded for some but not all p ∈ ω*.
3
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Two results on special points

100%
Dow A.
Fundamenta Mathematicae
|
2003
|
tom 176
|
nr 2
171-179
EN
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.
4
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More about spaces with a small diagonal

64%
Dow A., Pavlov O.
Fundamenta Mathematicae
|
2006
|
tom 191
|
nr 1
67-80
EN
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 that this holds in models of CH. In the present paper we prove that this also follows from the Proper Forcing Axiom (PFA). We furthermore present two (consistent) examples of countably compact non-metrizable spaces with small diagonal, one of which maps perfectly onto ω₁.
5
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Ordinal remainders of classical ψ-spaces

64%
Dow A., Vaughan J.
Fundamenta Mathematicae
|
2012
|
tom 217
|
nr 1
83-93
EN
Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain ${T_{α}: α < λ}$ of infinite subsets of ω, there exists $ℳ ⊂ [ω]^{ω}$, 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 < 𝔱⁺, where 𝔱 is the tower number, there exists a mod-finite ascending chain ${T_{α}: α < λ}$, hence a ψ-space with Stone-Čech remainder homeomorphic to λ +1. This generalizes a result credited to S. Mrówka by J. Terasawa which states that there is a MADF ℳ such that βψ∖ψ is homeomorphic to ω₁ + 1.
6
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The measure algebra does not always embed

64%
Dow A., Hart K. P.
Fundamenta Mathematicae
|
2000
|
tom 163
|
nr 2
163-176
EN
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.
7
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Two-to-one continuous images of ℕ*

64%
Dow A., Techanie G.
Fundamenta Mathematicae
|
2005
|
tom 186
|
nr 2
177-192
EN
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.
8
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Is 𝓟(ω) a subalgebra?

64%
Dow A., Farah I.
Fundamenta Mathematicae
|
2004
|
tom 183
|
nr 2
91-108
EN
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 βℕ if it contains a zero-set that maps onto βℕ.
9
Content available remote

More on tie-points and homeomorphism in ℕ*

64%
Dow A., Shelah S.
Fundamenta Mathematicae
|
2009
|
tom 203
|
nr 3
191-210
EN
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 $X = A {⋈ \limits_{x}} B$ 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 of an involution on ℕ*. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of ℕ* which is not a homeomorph of ℕ*.
10
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Representing free Boolean algebras

64%
Dow A., Nyikos P.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 1
21-30
EN
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 $ℵ_1$ generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra of any partitioner algebra.
11
Content available remote

Reflecting Lindelöf and converging ω₁-sequences

64%
Dow A., Hart K.
Fundamenta Mathematicae
|
2014
|
tom 224
|
nr 3
205-218
EN
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, has many ℵ₁-sized Lindelöf subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.
12
Content available remote

Are initially $ω_1$ -compact separable regular spaces compact?

64%
Dow A., Juhász I.
Fundamenta Mathematicae
|
1997
|
tom 154
|
nr 2
123-132
EN
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.
13
Content available remote

Remote points in spaces with π-weight $ω_1$

32%
Dow A.
Fundamenta Mathematicae
|
1984
|
tom 124
|
nr 3
197-205
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