REIN UW  Institutional Repository of the University of Warsaw

Żuławnik, Ludwik

Bławat, Magdalena

Jastrzębski, Jacek

Protection of purchasers rights in the development contract

Drosio, Katarzyna

Smoliński, Mariusz

Jurkowlaniec, Grażyna

The Work of Kazimierz Andrzej Niedbałowicz in Lviv at the Turn of the 17th and  18th Centuries

Lipiński, Krystian

Makowski, Jerzy

Kałaska, Maciej

The gateway to the Colca Canyon. Functional spatial structure and digital perception of Chivay in the years 1997-2025

Mazurek, Dominik

Ostaszewski, Wojciech

Gersdorf, Małgorzata

Expansion of labor law institutions into civil law employment

Puchalska, Michalina

Radzikowski, Krzysztof

Modzelewski, Witold

Assessed tax and the right of assessment in the Polish tax system

final_published2(27)Inicjatywa Doskonałości – Uczelnia Badawcz

Strupiechowska, Magdalena

"Job crafting" jako przykład ideologii nowego zarządzania – krytyczna analiza w optyce nowego ducha kapitalizmu Boltanskiego i Chiapello

Zakrzewski, Krzysztof

Marciszewski, Witold

Kalenda, Ondře

Borodulin-Nadzieja, Piotr

Kania, Tomasz

Year: 2025
Corson-like Compacta and Related Function Spaces
Zakrzewski, Krzysztof
Posted at The Institutional Repository of the University of Warsaw
ReIn UW: https://repozytorium.uw.edu.pl//handle/item/176989
Unique UUID of the publication: e4723955-8d06-409f-abdb-06ef1e63d6f
Abstract of doctoral thesis of Krzysztof Zakrzewski
Krzysztof Zakrzewski
24 April 2025
Compactness has always played an important role in analysis. One of the most well known classes
of compact spaces with roots in functional analysis is the class of Eberlein compact spaces, i.e. weakly
compact subsets of Banach spaces. An important extension of this class is the class of Corson compact
spaces, i.e. compact spaces that can be embedded into Σ-products of Euclidean lines (see [Ne] and
[AMN]). Both of these classes for decades have been the object of research by numerous mathematicians
from many research centers around the world (see [HNV]).
The thesis is mainly devoted to the study of the classes of κ-Corson compact spaces, which are a
generalization of the notion of Corson compact space (the class of Corson compacta coincides with the
class of ω1-Corson compacta), and a certain subclass of Eberlein compact spaces - the class of NY
compact spaces.
For arbitrary infinite cardinal number κ, κ-Corson compact spaces were first considered by Kalenda
[Ka2] and in [BM] (where the terminology was slightly different). Trying to examine them system-
atically one discovers quickly that the case κ = ω is quite special. In turn, for regular, uncountable
cardinal numbers κ, there is a number of properties of κ-Corson compacta that may be seen as natural
analogues of known features of the usual Corson compact spaces. In Sections 2.2 – 2.4 we investigate
ω-Corson compacta and a wider, somewhat more natural, class of NY compacta, first considered by
Nakhmanson and Yakovlev [NY].
In Chapter 2, extending results from [NY], we give internal characterizations of ω-Corson compacta
(Theorem 2.3.6) and of NY compacta (Theorem 2.3.1). We discuss the stability of these classes
of compacta under countable products (Corollary 2.4.2), continuous images (Corollary 2.4.3), and
the functor P which assigns to a compact space K the compact space P (K) of regular probability
Borel measures equipped with the weak∗ topology (Remark 2.4.11). In [NY] it was proved that all
NY compacta are hereditarily metacompact, in Section 2.4 we investigate this covering property for
Eberlein compacta not belonging to the class of NY compact spaces (Example 2.4.5, Theorem 2.4.9).
In Section 2.5 we show that the question, for which infinite cardinal numbers κ, the class of κ-
Corson compact spaces is closed under the functor P is naturally related to the notion of a caliber of
measures (so is, typically, undecidable in the usual set theory). Results presented in Chapter 2 come
from the article [MPZ].
In Chapter 3 we consider spaces of real continuous functions Cp(K) on some compact spaces K.
In the theory of function spaces there are many theorems answering questions of the following type:
Assume we are given spaces X and Y and a topological property P. Assume further that the function
spaces Cp(X) and Cp(Y ) are (linearly) homeomorphic. Is it true that if a space X has the property P,
then the space Y also must have this property ? There is a variety of topological properties for which
the answer to such question is in the positive. Somewhat better type of such theorems characterise
given topological property P of a space X in terms of properties of the function space Cp(X). In this
chapter we will present results of both types.
Section 3.2 contains a proof of Theorem 3.2.11 stating that, for σ-compact spaces X and Y , where
X is strongly countable dimensional, if function spaces Cp(X) and Cp(Y ) are linearly homeomorphic,
then Y is strongly countable dimensional as well. We also investigate the case when X is finite
dimensional (Theorem 3.2.12).
In Section 3.3 we show that the class of NY compact spaces is invariant under linear homeomor-
phisms of function spaces. In fact we prove several slightly more general theorems (Theorems 3.3.9,
3.3.11, 3.3.12). Recently these results were refined by Krupski and Avilés. They proved that the class
of NY compact spaces is preserved by homeomorphisms of function spaces [AK]. We also give an exam-
ple of a compact space K and a ω-Corson compact space L such that Cp(K) is linearly homeomorphic
to a linear subspace of Cp(L) but K is not Eberlein compact (Example 3.3.13).
1
Section 3.4 is devoted to the proof of a characterisation (Theorem 3.4.8) of κ-Corson compact
spaces K in terms of topological properties of function spaces Cp(K) for regular uncountable cardinal
numbers κ, which extends results of Pol [P] and Bell and Marciszewski [BM]. Additionally, using results
from Chapters 2 and 3 we show that the class of ω-Corson compact spaces K is invariant under linear
homeomorphisms of function spaces Cp(K) (Theorem 3.4.14).
In the final section of Chapter 3 we discuss the stability of κ-Corson compacta K with respect to
isomorphisms of Banach spaces C(K). Theorems discussed in this section come from [MPZ]. Results
presented in the rest of Chapter 3 come from the article [Za], and were partially announced and
discussed in [MPZ].
In the first part of the last chapter we discuss two results concerning continuous linear operators
between general locally convex, linear topological spaces. We show that every open operator is always
weakly open (Corollary 4.1.5). We also present a theorem generalising the well known Banach theorem
about closed transformations to a general setting of locally convex spaces (Theorem 4.1.7). In the
second part we show that c0-product of function spaces defined on a pseudocompact space is always
homeomorphic to its countable power (Theorem 4.2.3). This theorem generalises theorem of Gul’ko
and Khmyleva stating that the space c0 of sequences converging to 0, endowed with the pointwise
convergence topology, is homeomorphic to its countable power.
2


Przestrzenie zwarte podobne do przestrzeni zwartych Carsona oraz powiązane przestrzenie funkcyjne

Gancarz, Artur

Szafrański, Dariusz

Wiącek, Marcin

PUBLIC VERSUS PRIVATE THEATRE (PUBLIC LAW ASPECTS)

Wasylów, Aleksandra

Litwinowicz-Droździel, Małgorzata

Schollenberger, Justyna

The dimensions of the search for subjectivity in Wiesław Myśliwski’s The Palace

Zalewska, Weronika

Malesa, Katarzyna

Eberharter, Markus

The application of modern technologies in foreign language acquisition for individuals with visual impairment

Not a member yet