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Register a new userA lower density operator for the Borel algebra
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We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $sigma$-ideal of countable sets, for an uncountable Polish space, is equivalent to the Continuum Hypothesis.
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In recent years much attention has been enjoyed by topological spaces which are dominated by second countable spaces. The origin of the concept dates back to the 1979 paper of Talagrand in which it was shown that for a compact space X, Cp(X) is dominated by P, the set of irrationals, if and only if Cp(X) is K-analytic. Cascales extended this result to spaces X which are angelic and finally in 2005 Tkachuk proved that the Talagrand result is true for all Tychnoff spaces X. In recent years, the notion of P-domination has enjoyed attention independent of Cp(X). In particular, Cascales, Orihuela and Tkachuk proved that a Dieudonne complete space is K-analytic if and only if it is dominated by P. A notion related to P-domination is that of strong P- domination. Christensen had earlier shown that a second countable space is strongly P-dominated if and only if it is completely metrizable. We show that a very small modification of the definition of P-domination characterizes Borel subsets of Polish spaces.
This paper is a contribution to understanding what properties should a topological algebra on a Stone space satisfy to be profinite. We reformulate and simplify proofs for some known properties using syntactic congruences. We also clarify the role of various alternative ways of describing syntactic congruences, namely by finite sets of terms and by compact sets of continuous self mappings of the algebra.
As proved by Dimov [Acta Math. Hungarica, 129 (2010), 314--349], there exists a duality L between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight and of dimension of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X)=w(L(X)), and if, in addition, X is normal, then dim(X)=dim(L(X)).
We prove that the minimal left ideals of the superextension $lambda(Z)$ of the discrete group $Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $lambda(Z_2)$ of the compact group $Z_2$ of integer 2-adic numbers.
Hindmans celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Mengers classic covering property. The methods include, in addition to Hurewiczs game theoretic characterization of Mengers property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides stro
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