Extension operators on balls and on spaces of finite sets

We study extension operators between spaces $\sigma_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0<\lambda<\mu$, there is no extension operator $T: C(\lambda B_H)\to C(\mu B_H)$

On measures on Rosenthal compacta

We show that if K is Rosenthal compact which can be represented by functions with countably many discontinuities then every Radon measure on K is countably determined. We also present an alternative proof of the result stating that every Radon measure on an arbitrary Rosenthal compactum is of countable type. Our approach is based on some caliber-type properties of measures, parameterized by separable metrizable spaces.Comment: 14 page

Would Leibniz have shared von Neumann’s logical physicalism?

This paper represents such an amateur approach; hence any comments backed up by professional erudition will be highly appreciated. Let me start from an attempt to sketch a relationship between professionals’ and amateurs’ contributions. The latter may be compared with the letters to the Editor of a journal, written by perceptive readers, while professionals contribute to the very content of the journal in question. Owing to such letters, the Editor and his professional staff can become more aware of the responses of educated public to the journal’s output

P-filters and hereditary Baire function spaces

AbstractWe extend the results of Gul'ko and Sokolov proving that a filter F on ω, regarded as a subspace of the Cantor set 2ω, is a hereditary Baire space if and only if F is a nonmeager (i.e., second category) P-filter. We also prove related results on hereditary Baire spaces of continuous functions Cp(X)

The Josefson--Nissenzweig theorem and filters on ω\omega

For a free filter $F$ on $\omega$, endow the space $N_F=\omega\cup\{p_F\}$, where $p_F\not\in\omega$, with the topology in which every element of $\omega$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. Spaces of the form $N_F$ constitute the class of the simplest non-discrete Tychonoff spaces. In this paper we study them in the context of the celebrated Josefson--Nissenzweig theorem from Banach space theory, e.g., we completely describe those filters $F$ for which the spaces $N_F$ carry sequences $\langle\mu_n\colon n\in\omega\rangle$ of finitely supported signed measures satisfying the following two conditions: $\|\mu_n\|=1$ for every $n\in\omega$, and $\mu_n(f)\to 0$ for every bounded continuous real-valued function $f$ on $N_F$. As a consequence, we obtain a description of a wide class of filters $F$ having the following properties: (1) if $X$ is a Tychonoff space and $N_F$ is homeomorphic to a subspace of $X$, then the space $C_p^*(X)$ of bounded continuous real-valued functions on $X$ contains a complemented copy of the space $c_0$ endowed with the pointwise topology, (2) if $K$ is a compact Hausdorff space and $N_F$ is homeomorphic to a subspace of $K$, then the Banach space $C(K)$ of continuous real-valued functions on $K$ is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space $K$ contains a non-trivial convergent sequence, then the space $C(K)$ is not Grothendieck.Comment: 46 pages, comments are welcome

A countable dense homogeneous topological vector space is a Baire space

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru\v{s}\'ak, Zamora Avil\'es, and Hern\'andez-Guti\'errez concerning countable dense homogeneous products.Comment: slightly modified and expanded versio