264 research outputs found
A new Lindelof topological group
We show that the subsemigroup of the product of w_1-many circles generated by
the L-space constructed by J. Moore is again an L-space. This leads to a new
example of a Lindelof topological group. The question whether all finite powers
of this group are Lindelof remains open
CH, a problem of Rolewicz and bidiscrete systems
We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable āniceā semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelƶf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum. We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K2 which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that for every infinite compact Hausdorff space K, the space C(K) has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K
Finitely fibered Rosenthal compacta and trees
We study some topological properties of trees with the interval topology. In
particular, we characterize trees which admit a 2-fibered compactification and
we present two examples of trees whose one-point compactifications are
Rosenthal compact with certain renorming properties of their spaces of
continuous functions.Comment: Small changes, mainly in the introduction and in final remark
On some classes of Lindel\"of Sigma-spaces
We consider special subclasses of the class of Lindel\"of Sigma-spaces
obtained by imposing restrictions on the weight of the elements of compact
covers that admit countable networks: A space is in the class
if it admits a cover by compact subspaces of weight
and a countable network for the cover. We restrict our attention to
. In the case , the class includes the class
of metrizably fibered spaces considered by Tkachuk, and the -approximable
spaces considered by Tkacenko. The case corresponds to the spaces of
countable network weight, but even the case gives rise to a
nontrivial class of spaces. The relation of known classes of compact spaces to
these classes is considered. It is shown that not every Corson compact of
weight is in the class , answering a question
of Tkachuk. As well, we study whether certain compact spaces in
have dense metrizable subspaces, partially answering a
question of Tkacenko. Other interesting results and examples are obtained, and
we conclude the paper with a number of open questions.Comment: 21 pages. to appear in Topology and its Application
Trees and EhrenfeuchtāFraıĢssĆ© games
AbstractTrees are natural generalizations of ordinals and this is especially apparent when one tries to find an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals
Separating club-guessing principles in the presence of fat forcing axioms
We separate various weak forms of Club Guessing at in the presence of large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with -sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds -many reals but preserves CH
Some Banach spaces added by a Cohen real
We study certain Banach spaces that are added in the extension by one Cohen
real. Specifically, we show that adding just one Cohen real to any model adds a
Banach space of density which does not embed into any such space in
the ground model such a Banach space can be chosen to be UG This has
consequences on the the isomorphic universality number for Banach spaces of
density , which is hence equal to in the standard Cohen
model and the same is true for UG spaces. Analogous universality results for
Banach spaces are true for other cardinals, by a different proof.Comment: The version to appear in Topology and Its Applications arXiv admin
note: substantial text overlap with arXiv:1308.364
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