46 research outputs found
Extension operators on balls and on spaces of finite sets
We study extension operators between spaces of subsets of
of cardinality at most . As an application, we show that if is the
unit ball of a nonseparable Hilbert space , equipped with the weak topology,
then, for any , there is no extension operator
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
For a free filter on , endow the space ,
where , with the topology in which every element of
is isolated whereas all open neighborhoods of are of the form
for . Spaces of the form 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 for which the spaces
carry sequences of finitely
supported signed measures satisfying the following two conditions:
for every , and for every bounded
continuous real-valued function on . As a consequence, we obtain a
description of a wide class of filters having the following properties: (1)
if is a Tychonoff space and is homeomorphic to a subspace of ,
then the space of bounded continuous real-valued functions on
contains a complemented copy of the space endowed with the pointwise
topology, (2) if is a compact Hausdorff space and is homeomorphic to
a subspace of , then the Banach space of continuous real-valued
functions on is not a Grothendieck space. The latter result generalizes the
well-known fact stating that if a compact Hausdorff space contains a
non-trivial convergent sequence, then the space 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 , the function space
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 (dual Banach space ), the space
equipped with the weak topology ( with the weak 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