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Register a new userOn fragmentable compact lines
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We prove that if a compact line is fragmentable, then it is a Radon-Nikodym compact space.
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We use lexicographic products to give examples of compact spaces of first Baire class functions on a compact metric space that cannot be represented as spaces of functions with countably many discontinuities.
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.
In this paper, we intend to show that under not too restrictive conditions, results much stronger than the one obtained earlier by Hejduk could be established in category bases.
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.
As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,alpha])$, where $alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,alpha]) to C_p(Y)$ is given. We also observe that for every countable ordinal $alpha$ all closed linear subspaces of $C_p([1,alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X subset mathbb{R}^n$).
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