Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Haar meager sets
132
0
0.0
(
0
)
Authors
U. B. Darji
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We show that an ideal $mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that ${n_1,n_2,ldots,} otin mathcal{I}$). The analogous characterization holds also for $mathcal{I}$-cluster points.
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.
For a non-isolated point $x$ of a topological space $X$ the network character $nw_chi(x)$ is the smallest cardinality of a family of infinite subsets of $X$ such that each neighborhood $O(x)$ of $x$ contains a set from the family. We prove that (1) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $nw_chi(x)=aleph_0$; (2) for each point $xin X$ with countable character there is an injective sequence in $X$ that $F$-converges to $x$ for some meager filter $F$ on $omega$; (3) if a functionally Hausdorff space $X$ contains an $F$-convergent injective sequence for some meager filter $F$, then for every $T_1$-space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. Also we investigate properties of filters $F$ admitting an injective $F$-convergent sequence in $betaomega$.
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.
We prove that if a compact line is fragmentable, then it is a Radon-Nikodym compact space.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
