Loading [MathJax]/extensions/MathZoom.js
Displaying 141 –
160 of
332
We deal with Boolean algebras and their cardinal functions: π-weight π and π-character πχ. We investigate the spectrum of π-weights of subalgebras of a Boolean algebra B. Next we show that the π-character of an ultraproduct of Boolean algebras may be different from the ultraproduct of the π-characters of the factors.
Let be a Polish ideal space and let be any set. We show that under some conditions on a relation it is possible to find a set such that is completely -nonmeasurable, i.e, it is -nonmeasurable in every positive Borel set. We also obtain such a set simultaneously for continuum many relations Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.
A function f: ℝ → {0,1} is weakly symmetric (resp. weakly symmetrically continuous) at x ∈ ℝ provided there is a sequence hₙ → 0 such that f(x+hₙ) = f(x-hₙ) = f(x) (resp. f(x+hₙ) = f(x-hₙ)) for every n. We characterize the sets S(f) of all points at which f fails to be weakly symmetrically continuous and show that f must be weakly symmetric at some x ∈ ℝ∖S(f). In particular, there is no f: ℝ → {0,1} which is nowhere weakly symmetric.
It is also shown that if at each point x we...
We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers ℙ not and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ ℙ of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. The answer to the question as stated is negative if X is Baire but Y is not Baire.
Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where is non-well-ordered. One of the interesting properties of this number is that the Hanf number of is exactly . It was proved in [BK71] that if ℵ₀ < λ < κ2λ = κ∙ ;
∙ cf(θ) ≥ λ⁺ and whenever μ < θ;
∙ .
Then there is a forcing...
It is shown that a space is -Weakly Fréchet-Urysohn for iff it is -Weakly Fréchet-Urysohn for arbitrary , where is the -th left power of and for . We also prove that for -compact spaces, -sequentiality and the property of being a -Weakly Fréchet-Urysohn space with , are equivalent; consequently if is -compact and , then is -sequential iff is -sequential (Boldjiev and Malyhin gave, for each -point , an example of a compact space which is -Fréchet-Urysohn and it is...
We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-ℵ₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question...
We consider a set, L, of lines in and a partition of L into some number of sets: . We seek a corresponding partition such that each line l in meets the set in a set whose cardinality has some fixed bound, . We determine equivalences between the bounds on the size of the continuum, , and some relationships between p, and .
We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space whose square is again Lindelöf but its cube has a closed discrete subspace of size , hence the Lindelöf degree . In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space such that for all positive integers , but .
Currently displaying 141 –
160 of
332