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Displaying 301 –
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436
By an - tree we mean a tree of power and height . Under CH and we call an -tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between and . In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus that there only exist Kurepa trees with -many branches, which answers another...
If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. The cardinal invariant A(G) is defined to be the least uncountable cardinal in the abelian subgroup spectrum of G. The value of A(G) is examined for various groups G which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the...
We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees.
...
Module is said to be small if it is not
a union of strictly increasing infinite
countable chain of submodules. We show
that the class of all small modules
over self-injective purely infinite
ring is closed under direct products
whenever there exists no strongly
inaccessible cardinal.
We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and “all Aronszajn trees are special” is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from [“Proper and Improper Forcing”, Ch. V]) and < ω₁-proper -bounding forcings adding reals. We show...
Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift...
Assuming ⋄, we construct a connected compact topological space K such that for every closed L ⊂ K the Banach space C(L) has few operators, in the sense that every operator on C(L) is multiplication by a continuous function plus a weakly compact operator. In particular, C(K) is indecomposable and has continuum many non-isomorphic indecomposable quotients, and K does not contain a homeomorphic copy of βℕ.
Moreover, assuming CH we construct a connected compact K where C(K) has few...
We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...
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,...
We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies Δ(X) > e(X) is ω-resolvable. Here Δ(X), the dispersion character of X, is the smallest size of a non-empty open set in X, and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable.
We also prove that any regular...
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