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Register a new userAnother characterization of meager ideals
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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.
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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.
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 $betaomega$.
A {it weak selection} on $mathbb{R}$ is a function $f: [mathbb{R}]^2 to mathbb{R}$ such that $f({x,y}) in {x,y}$ for each ${x,y} in [mathbb{R}]^2$. In this article, we continue with the study (which was initiated in cite{ag}) of the outer measures $lambda_f$ on the real line $mathbb{R}$ defined by weak selections $f$. One of the main results is to show that $CH$ is equivalent to the existence of a weak selection $f$ for which: [ mathcal lambda_f(A)= begin{cases} 0 & text{if $|A| leq omega$,} infty & text{otherwise.} end{cases} ] Some conditions are given for a $sigma$-ideal of $mathbb{R}$ in order to be exactly the family $mathcal{N}_f$ of $lambda_f$-null subsets for some weak selection $f$. It is shown that there are $2^mathfrak{c}$ pairwise distinct ideals on $mathbb{R}$ of the form $mathcal{N}_f$, where $f$ is a weak selection. Also we prove that Martin Axiom implies the existence of a weak selection $f$ such that $mathcal{N}_f$ is exactly the $sigma$-ideal of meager subsets of $mathbb{R}$. Finally, we shall study pairs of weak selections which are almost equal but they have different families of $lambda_f$-measurable sets.
We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $Delta$-space in the sense of cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a v{C}ech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mrowka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,omega_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $Delta$-spaces is invariant under basic topological operations.
One of the main obstacle to study compactness in topological spaces via ideals was the definition of ideal convergence of subsequences as in the existing literature according to which subsequence of an ideal convergent sequence may fail to be ideal convergent with respect to same ideal. This obstacle has been get removed in this article and notions of I compactness as well as I star compactness of topological spaces have been introduced and studied to some extent. Involvement of I nonthin subsequences in the definition of I and I star compactness make them different from compactness even in metric spaces.
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