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A Corson compact L-space from a Suslin tree

Peter Nyikos (2015)

Colloquium Mathematicae

The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

Almost maximal topologies on groups

Yevhen Zelenyuk (2016)

Fundamenta Mathematicae

Let G be a countably infinite group. We show that for every finite absolute coretract S, there is a regular left invariant topology on G whose ultrafilter semigroup is isomorphic to S. As consequences we prove that (1) there is a right maximal idempotent in βG∖G which is not strongly right maximal, and (2) for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination (-,-,+), there is a corresponding regular almost maximal left invariant...

Asymmetric tie-points and almost clopen subsets of *

Alan S. Dow, Saharon Shelah (2018)

Commentationes Mathematicae Universitatis Carolinae

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...

Cardinal invariants of universals

Gareth Fairey, Paul Gartside, Andrew Marsh (2005)

Commentationes Mathematicae Universitatis Carolinae

We examine when a space X has a zero set universal parametrised by a metrisable space of minimal weight and show that this depends on the σ -weight of X when X is perfectly normal. We also show that if Y parametrises a zero set universal for X then h L ( X n ) h d ( Y ) for all n . We construct zero set universals that have nice properties (such as separability or ccc) in the case where the space has a K -coarser topology. Examples are given including an S space with zero set universal parametrised by an L space (and...

Coronas of ultrametric spaces

Igor V. Protasov (2011)

Commentationes Mathematicae Universitatis Carolinae

We show that, under CH, the corona of a countable ultrametric space is homeomorphic to ω * . As a corollary, we get the same statements for the Higson’s corona of a proper ultrametric space and the space of ends of a countable locally finite group.

Countable Compact Scattered T₂ Spaces and Weak Forms of AC

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

Countable dense homogeneous filters and the Menger covering property

Dušan Repovš, Lyubomyr Zdomskyy, Shuguo Zhang (2014)

Fundamenta Mathematicae

We present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain for any n ∈ ω ∪ {∞} an n-dimensional metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.

Extending real-valued functions in βκ

Alan Dow (1997)

Fundamenta Mathematicae

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 c and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.

Extending the ideal of nowhere dense subsets of rationals to a P-ideal

Rafał Filipów, Nikodem Mrożek, Ireneusz Recław, Piotr Szuca (2013)

Commentationes Mathematicae Universitatis Carolinae

We show that the ideal of nowhere dense subsets of rationals cannot be extended to an analytic P-ideal, F σ ideal nor maximal P-ideal. We also consider a problem of extendability to a non-meager P-ideals (in particular, to maximal P-ideals).

Further results on neutral consensus functions

G. D. Crown, M.-F. Janowitz, R. C. Powers (1995)

Mathématiques et Sciences Humaines

We use a set theoretic approach to consensus by viewing an object as a set of smaller pieces called “bricks”. A consensus function is neutral if there exists a family D of sets such that a brick s is in the output of a profile if and only if the set of positions with objects that contain s belongs to D. We give sufficient set theoretic conditions for D to be a lattice filter and, in the case of a finite lattice, these conditions turn out to be necessary. Ourfinal result, which involves a finite...

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