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

The Problem of Modern Monetization of Memes: How Copyright Law Can Give Protection to Meme Creators

Some legal questions answered in this article on the horizon for the courts and lawyers is how should courts apply copyright law to popular media made by small scale creators and shared on the internet, otherwise known as memes. Part II of this article will focus on validity of potential copyright protection in internet memes. It will start by describing the increased monetization surrounding memes and how this monetization calls for greater interest for meme creators to protect their work. It will then describe the merits of individual copyright interests in internet memes. Part III of this article will focus on how memes have existed without copyright lawsuits from content creators: principally, that internet memes constitute fair use. This section will use an example meme to weigh all four statutory factors of fair use to support the argument that internet memes are highly transformative and do not impact the market of the original copyrighted work. Part IV of this article will outline how public policy favors copyright protection of memes since copyright protection would not stifle creativity or new meme creations. First, copyright protection of memes would not disrupt the current “meme culture” of sharing memes because social media platforms, the major platform and vehicle for meme creation and sharing, have negated many copyright concerns through their terms of use policies. Next, it will explain how the Digital Media Copyright Act’s safe harbor rule protects social media platforms from being secondarily liable for potential copyright infringements involving meme appropriation. Finally, it will explain how other aspects of copyright law, like independent creation, the idea/expression dichotomy, and the fair use doctrine, will prevent meme creators from “weaponizing” their copyright interests in their memes

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)

Function spaces on τ\tau-Corson compacta and tightness of polyadic spaces

summary:We apply the general theory of $\tau$-Corson Compact spaces to remove an unnecessary hypothesis of zero-dimensionality from a theorem on polyadic spaces of tightness $\tau$. In particular, we prove that polyadic spaces of countable tightness are Uniform Eberlein compact spaces

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