Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userP-domination and Borel sets
187
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
The notion of Haar null set was introduced by J. P. R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke. During the last twenty years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. Inspired by these various results, we introduce the topological analogue of the notion of Haar null set. We call it Haar meager set. We prove some basic properties of this notion, state some open problems and suggest a possible line of investigation which may lead to the unification of these two notions in certain context.
We study products of general topological spaces with Mengers covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups $C(X,bbR)$ of continuous real-valued functions on complete metric spaces $X$, with respect to the compact-open topology.
An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduce the problem to infinite metrizable profinite groups. We provide a positive answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
