Topologia

The space in question is the space $\textfrak M$ of Lebesgue measurable subsets of the unit interval equipped with the usual Fr'echet--Nikodym (semi)metric. We show that there exists a sequence of elements of $\textfrak M$ such that their mutual distances are > 1/2. It seems to be an open problem whether "1/2" can be replaced here by a bigger constant C. We show that C must be smaller than 9/14. Moreover, we present a version of the problem in terms of binary codes.

We present a construction of a large family of Borel linear subspaces of $\mathbb{R}$ (over $\mathbb{Q}$), which are incomparable with respect to Borel linear embeddings (over $\mathbb{Q}$). A version of this construction answers a question by Frisch and Shinko.

Combinatorial space is a type of Banach space induced by (some) family of sets in a certain way. During the talk I will present examples of families (of subsets of natural numbers) and spaces related with them. Furthermore, I will consider properties of these spaces, in particular in the context of containing isomorphic copy of c_0 and \ell_1.

We will discuss adding Cohen reals for various types of trees on Baire and Cantor space. We will distinguish that these Cohen reals can be added in a 'strong' or 'weak' way. While the former has rather pathological consequences, the latter allows certain control over the ideal related to the tree type.

We will consider examples of analytic sets which are not Borel. We will focus on, so called, complete analytic sets. Firstly, we will consider ideals on naturals (naturally treated as subsets of the Cantor space). Secondly, we will consider the family of Silver trees. We will compare the later example with theorem of Kechris-Louveau-Woodin.

I will present an intersection theorem for a descending sequence of closed sets for which a convexity type condition is satisfied. However, the condition applies to the whole sequence, not separately to individual sets. I will show that the following theorems follows easily from this result: Smulian's theorem about characterization of reflexive spaces, theorem about the convex set in Hilbert space, and Browder-Gohde-Kirk's theorem about fixed points of nonexpansive mappings.

In this talk, we will continue the construction of a Michael space from an ultrafilter. We will show the consistency of ZFC + There are no Michael Ultrafilters and we will discuss some open questions.

(join work with A. Illanes, V. Martı́nez-de-la-Vega, and J. M. Martı́nez-Montejano) Bobok, Pyrih and Vejnar presented six kinds of blocking properties for points in continua. We can consider the same properties for subcontinua. During my talk I will present some results concerning the blocking properties of the diagonal in Cartesian product. Among others, I will show a new characterisation of the interval.

P-points are ultrafilters in which every decreasing sequence of sets from the filter has a pseudointersection (in a sense an intersection modulo finite sets) in the filter. Quite similarly P-measures (known in the literature as measures with additive property*) are finitely additive probability measures in which every decreasing sequence of sets has a pseudointersection with measure equal to the limit of measures of sets from the sequence. It is not hard to see that (a characteristic function of) a P-point is a P-measure. However, a question whether the existence of P-measures implies the existence of P-points remains open. I will talk about current knowledge of the problem including my and Piotr Borodulin-Nadzieja's efforts and results - based on the Silver forcing and its variations.

The Vietoris hyperspace of simple closed curves in the plane will be discussed toward its desirable characterization. In particular, the local contractibility will be shown. Joint work with Krzysztof Omiljanowski.

A family $\mathscr{I}\subseteq[\omega]^\omega$ is called ideal independent if no element $A\in\mathscr{I}$ is almost contained in the union of finitely many other elements in $\mathscr{I}$. The ideal independence number, denoted by $\mathfrak{s}{mm}$, is defined as the minimal cardinality of a maximal ideal independent family. We will review some results about ideal independent families and the cardinal invariant $\mathfrak{s}{mm}$. In particular we will prove that the ultrafilter number is a lower bound for $\mathfra{s}{mm}$. Also, we will see that the spectrum of ideal independent families, defined as the family of all cardinalities of maximal ideal independent families, can be quite rich. If time allows, we will sketch a proof that consistently $\mathfrak{s}{mm}<\mathfrak{a}_T$, where $\mathfrak{a}_T$ is the minimal cardinality of a family of disjoint compact sets covering the Baire space. This is joint work with V. Fischer and C. B. Switzer.

Preservation of reaping families and especially ultrafilters on countable sets is a well studied theme in set theory of the reals. A. Miller proved that if an ultrafilter remains a reaping family in some forcing extension, then it has to be also Sacks indestructible. The existence of Sacks indestructible ultrafilters in ZFC is an open question. A related problem is Sacks indestructibility of reaping families which are complements of ideals. We prove that complements of most classical ideals are indestructible with one notable exception, the ideal of sets asymptotic density zero. The presented results are from an upcoming paper with O. Guzman and M. Hrusak. About 15 minutes before the seminar we invite you for coffee and a chat.

Cembranos, Freniche, and Khurana (all independently) proved that for every two infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. To obtain this copy all the three proofs utilize in some way the Josefson--Nissenzweig theorem which more or less asserts that there is a sequence (mu_n) of normalized signed Radon measures on K\times L such that mu_n(f) converges to 0 for every f from C(K\times L). Since most (if not all) of the known proofs of the J--N theorem are non-constructive, it follows that the (known to me) proofs of Cembranos et al. are also non-constructive. During my talk I'll show a generalization of the theorem of Cembranos et al. whose proof uses a direct construction of a sequence (mu_n) of finitely supported measures on K\times L as above. I'll also discuss the case of pseudocompact spaces K and L and pose some questions.

We investigate Marczewski style ideals associated with the product of two tree-like forcing notions and compare these to original, one dimensional ones. Before the seminar we invite you for coffee and a chat.

Irwin and Solecki introduced projective Fraïssé theory and showed that the Fraïssé limit of the projective class of finite linear graphs is a pre-space of the pseudo-arc. This allowed to characterize the pseudo-arc as the unique approximatively projectively homogeneous arc-like continuum. We introduce a framework for Fraïssé theory where the pseudo-arc itself is a Fraïssé limit, and apply the framework to obtain similar characterizations for P-adic pseudo-solenoids. This is joint work with Wiesław Kubiś. About 15 minutes before the seminar we invite you for coffee and a chat.

We will discuss possible variants and generalizations of Eggleston theorem about inscribing big rectangles into big subsets of the plane. We will focus mainly on product of two Cantor spaces and comeager sets.

Given a family of real functions F we say that a set M ⊆ ℝ is magic for F if for all f, g ∈ F we have f [M ] ⊆ g[M ] ⇒ f = g. This notion was introduced by Diamond, Pomerance and Rubel in 1981 [1]. Recently some results about magic sets were proved by Halbeisen, Lischka and Schumacher [2]. Inspired by their work I constructed two families of magic sets one of them being almost disjoint and the other one being independent. During my talk I will sketch the background and present the proof for the almost disjoint family, which uses a Kurepa tree.

We present a construction of two separable compacta K and L such that C(L) is a direct sum of C(K) and some Banach space X which is not isomorphic to a space of continuous functions. Joint work with Alberto Salguero Alarcon (Badajoz).

A Lindelöf space X is Michael if it has a non-Lindelöf product with the space of irrational numbers. The existence of these kinds of spaces using only the standard axioms of ZFC is still unknown. We will look into some of the combinatorics related to this problem and discuss its relationships with Selective Ultrafilters.

We consider compact spaces which can be embedded into a product of real lines so that the support of every element is of size < kappa; here kappa is a fixed cardinal number. We discuss measure-theoretic properties of such spaces and related properties of Banach spaces of continuous functions.

A twisted sum of two Banach spaces X and Y is another space Z containing Y as a closed subspace so that Z/Y = X. In this talk we focus on twisted sums of spaces of continuous functions on compact spaces. It has been known for some time that a twisted sum of two C(K)-spaces does not need to be isomorphic to a C(K)-space (see for example [1, theorem 3.5.b] or [2]). We will focus on one recent and singular construction which serves as an example: a twisted sum of c0 and c0 (c) which is not isomorphic to any C(K)-space. This is part of a joint work with Grzegorz Plebanek.

There are deep connections between model theory of the infinitary logic and descriptive set theory: Scott analysis, the López-Escobar theorem or the Suzuki theorem are well known examples of this phenomenon. In this talk, I will present results of a research devoted to generalizing these connections to the setting of continuous infinitary logic and Polish metric structures. In particular, I will discuss a continuous counterpart of a theorem of Hjorth and Kechris characterizing essential countability of the isomorphism relation on a given Borel class of countable structures. As an application, I will give a short model-theoretic proof of a result of Kechris saying that orbit equivalence relations induced by continuous actions of locally compact Polish groups are essentially countable. This is joint work with Andreas Hallbäck and Todor Tsankov.

Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory T , T_∃∨∀ denotes the logical consequences of T which are boolean combinations of universal sentences. T∗ is the AMC of T if it is model complete and T_∃∨∀ = T*_∃∨∀. The {+, ·, 0, 1}-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as ∃x(x2 + 1 = 0) ∈ ACF_∃∨∀ \ Fields_∃∨∀. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) 2^aleph_0 = aleph_2 is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the ∈-theory ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently stronger notion of consistency for any mathematical problem ψ expressible as a Pi_2 -sentence of a (very large fragment of) third order arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free groups, are a small sample of such problems ψ). Partial Morleyizations can be described as follows: let Form_τ be the set of first order τ -formulae; for a subset A of Form_τ , τ_A is the expansion of τ adding atomic relation symbols R_φ for all formulae φ in A and T_τ,A is the τ_A -theory asserting that each τ -formula φ(x) ∈ A is logically equivalent to the corresponding atomic formula R_φ (x~x). For a τ -theory T T + Ti_τ,A is the partial Morleyization of T induced by A ⊆ F_τ.

Using a game-theoretic approach (Set-Cover game) we obtain a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We will present applications of this result to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.

Let us recall that a compact space K is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology. Our talk will be devoted to the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. Several such spaces were obtained using some additional set-theoretic assumptions. Recently, P. Koszmider constructed the first such example in ZFC. We investigate this problem for the class of Eberlein compact spaces. We construct such Eberlein compacta, assuming the existence of a Luzin set. We also show that it is consistent with ZFC that each Eberlein compact space of weight greater than $\omega_1$ contains a nonmetrizable closed zero-dimensional subspace. The talk is based on the paper "On two problems concerning Eberlein compacta":

D. Fremlin in 1975 has proved that if (X,S,m) is a probability space, then a sequence of measurable functions on X either has a subsequence convergent a.e., or there exists a subsequence without measurable pointwise cluster point. His proof is based upon the properties of weak convergent sequences in square integrable functions. The weaker form of the theorem was proved by Bucchioni and Goldman in1978. Their proof uses only some properties of the pair (family of measurable subsets of [0,1], family of null sets). The pair (family of subsets of [0,1] having the Baire property, family of sets of the first category) behaves similarly, so it was possible to obtain similar result for the convergence in category considered by E. Wagner in 1978. Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.

Shoenfield's celebrated absoluteness theorem says that no Sigma^1_2 fact phi can be shown to be independent of the axioms of ZFC via the method of forcing. A set of reals is universally Baire if its continuous preimages have the Baire property in all topological spaces. Can there be independence results about such sets? Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object g, let Gamma^infty_g be the set of universally Baire sets of V[g] and R_g be the set of reals of V[g]. Sealing (essentially) says that for all V-generic g and all V[g]-generic h there is an embedding j: L(Gamma^infty_g, R_g)-> L(Gamma^infty_g*h, R_g*h) Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem. It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing 2^{2^delta} to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem.

We consider interesting descriptive set-theoretic problems emerging from theoretical economics. First, we investigate a certain two-player game coming from gambling theory. Then, as a by-product, we obtain a novel game that characterises the Baire class 1 functions. Finally, we determine the exact complexity of the so-called value of the above game, which turns out to be a less well-known class, namely analytic-inductive.

We shall introduce a consistent set-theoretic axiom IIA which has a profound impact on convergence properties in topological groups. As an application we show that consistently (consequence of IIA) every countable sequential group is either metrizable or $k_\omega$.

We continue study of projective Fra\"{\i}ss\'e limit developed by Irvin, Panagiotopoulos and Solecki. We modify the ideas of monotone, confluent, or retraction from continuum theory as well as several properties of continua so as to apply to topological graphs. As the topological realizations of the Fra\"{\i}ss\'e limits we obtain either some known continua, for example the dendrite $D_3$ or the Cantor fan, or quite new, interesting ones for which we do not yet have topological characterizations. This is a joint work with Robert P. Roe.

Answering a question posed by R. Adikari and W. Lewis, I shall prove that for every zero-dimensional separable metric space G there is a Suslinian chainable continuum whose end points form a set homeomorphic to G. I will discuss some structural properties of such continua.

Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht--Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wide Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1). In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree. The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version DOI: 10.1017/jsl.2020.42

Given two measures that coincide on the intersection of their domains, can we find a measure that is a common extension of those two? Kellerer's results on marginal measures constitute an important partial positive answer. We will see how this is connected to some basic properties of the category of Banach lattices, like amalgamation and existence of injective objects. Joint work with Pedro Tradacete.

In this talk, I examine the following trichotomy which holds under the requirement that a sufficiently strong natural reflection principle should hold: The continuum ($=2^{\aleph_0}$) is either 1. $\aleph_1$ or 2. $\aleph_2$ or 3. fairly large. Here, the fair largeness of the continuum can be expressed either in terms of weak mahloness and/or some other ``large'' cardinal notions compatible with the continuum, or even in terms of existence of some saturated ideals. The reflection principles we consider here can be formulated as the following type of Downward Löwenheim-Skolem Theorems: 1'. For any structure $A$ of countable signature, there is an elementary substructure $B$ of $A$ of cardinality $<\aleph_2$ in terms of stationary logic. 2'. For any structure $A$ of countable signature, there is an elementary substructure $B$ of $A$ of cardinality $<2^{\aleph_0}$ in terms of stationary logic but only for formulas without free second order variables. 3'. For any structure $A$ of countable signature, there is an elementary substructure $B$ of $A$ of cardinality $<2^{\aleph_0}$ in terms of PKL logic (a variant of the stationary logic) in weak interpretation. The reflection points $<\aleph_2$ and $<2^{\aleph_0}$ can be considered to be natural/necessary since the reflection down to $<\aleph_2$ declares that $\aleph_1$ strongly represents the situation of uncountability; the reflection down to $<2^{\aleph_0}$ can be interpreted in the way that the reflection manifests that the continuum is very "rich". The Downward Löwenheim-Skolem Theorems in terms of stationary logics can be also regarded as very natural principles: They can be characterized in terms of Diagonal Reflection Principles of Sean Cox. Analyzing these three scenarios, we obtain the notion of Laver-generically large cardinals. Existence of a Laver-generically supercompact cardinal 1''. for $\sigma$-closed pos implies 1'.; 2''. for proper pos implies 2'.; while the existence of a Laver-generically supercompact cardinal 3''. for ccc pos implies 3'. The symmetry of the arguments involved suggests the possibility that the trichotomy might be a set-theoretic multiversal necessity. If time allows, I shall also discuss about the reflection of non-metrizability of topological spaces, Rado's Conjecture and Galvin's Conjecture in connection with the reflection properties in 1., 2. and 3.

A mathematical structure is called homogeneous if every isomorphism between its small substructures extends to an automorphism. Typically, ``small" means ``finite" or ``finitely generated". A stronger variant, which we call ``uniform homogeneity" requires that for each small substructure there is a suitable extension operator. We shall present examples of homogeneous but uniformly homogeneous structures. The talk is based on two works: one joint with S. Shelah (https://arxiv.org/abs/1811.09650), another one joint with B. Kuzeljevic (https://arxiv.org/abs/2004.13643).

This talk will be devoted to (products of) Menger and Hurewicz spaces and their connections to forcing and mad families. In particular, we shall show that in the Laver model, each mad family can be destroyed by a ccc poset preserving the ground model reals unbounded and splitting. It is an important open problem whether the same follows from CH.

There exist a natural forcing notion which turns given countable set into a Fraisse limit of a given Fraisse class. This long-known phenomenon provided a rough intuition that Fraisse limits, as "generic structures", have some connections with forcing. The goal of the talk is to look at some particular instances and possible applications of this idea.

We are going to establish basic properties of diagonal version of Silver forcing. Such forcing consists of partial functions p:\omega\rightarrow\omega with infinite codomain and p(n)<=n for each n\in dom(p). Cardinal characteristics of continuum will be calculated.

Given an almost disjoint family A, we consider K_A, the compact space defined by A. We discuss the number of nonisomorphic Banach spaces of the form C(K_A), of continuous functions.

A twisted sum of Banach spaces $X$ and $Y$ is another space $Z$ containing $Y$ as a subspace such that $Z/Y = X$. In this talk we study the behaviour of twisted sums in which $X$ is a $C(K)$-space, using techniques from Topology and Functional Analysis. Most of the action will take place in the space of continuous functions on the Cantor space.

For a countable cover $\mathcal{A}$ of a compact (Hausdorff) space $Y$ with closed subsets we define its height, which is a measure of its complexity and generalizes the notion of the Cantor-Bendixson rank. If $X$ is another compact space and $f : X \to Y$ is continuous, the cover can be pulled back to $X$ and its height may drop, but can never increase. We inspect how much the height can be reduced as $Y$ and $\mathcal{A}$ are fixed while $X$ and $f$ vary.

In the literature there are many different notions of porosity and one of them is the notion of strong porosity: Given a completely metrizable space X, a subset A of X is a strongly porous set if there is a positive constant p such that for any open ball B of radius r smaller than 1, there is an open ball B' inside of B of radius rp such that B' evades the set A. In this talk we will study the cardinal invariants of the sigma ideal generated by the strongly porous sets of the real line and the Cantor space, their relation with different notions of porosity and with other notions in set theory. In particular we will show the notion of strong porosity induces a combinatorial property on trees and we will see that these trees have a connection with the Sacks forcing and some cardinal invariants related to Martin's axiom.

Przestrzen topologiczna X jest Baire'a jesli dowolny przeliczalny przekroj zbiorow otwartych i gestych w X jest gesty w X. Przestrzen X jest dziedzicznie Baire'a jesli kazda domknieta podprzestrzen przestrzeni X jest Baire'a. Niech X bedzie przestrzenia metryczna i osrodkowa. W swoim odczycie zajme sie nastepujacym dosc naturalnym pytaniem: Jaka wlasnosc przestrzeni X jest rownowazna dziedzicznej wlasnosci Baire'a hiperprzestrzeni K(X) niepustych zwartych podzbiorow X, z metryka Hausdorffa. Niedawno Gartside, Medini i Zdomskyy podali odpowiedz na powyzsze pytanie i zauwazyli jego zwiazki z innymi naturalnymi pytaniami dotyczacymi wlasnosci narostow uzwarcen przestrzeni topologicznych. Pokaze jak twierdzenie Gartside'a, Mediniego i Zdomskyy'ego ma sie do pewnego twierdzenia Telgarsky'ego i naszkicuje alternatywny, bardziej elementarny dowod udowodnionego przez nich faktu.

Na odczycie przedstawie m. in. nowe przyklady analitycznych P-idealow inspirowane pewnymi przestrzeniami Banacha, jak i nowe (?) przyklady przestrzeni Banacha inspirowane pewnymi analitycznymi P-idealami.

We extend the theory of Borel/analytic equivalence relations and reductions between them to the theory of Borel/analytic pseudometrics and reductions between them. This is in the spirit of model theory for metric structures which aims to generalize discrete notions to their continuous counterparts. We consider several classical distances from functional analysis and metric geometry, such as Banach-Mazur distance, Gromov-Hausdorff distance, Kadets distance, Lipschitz distance, etc., and show how they reduce to each other in a Borel way. It is joint work with Marek Cúth and Ondřej Kurka.

Było to pytanie o to czy, w szczególności, produkt dwóch zwartych przestrzeni liniowo uporządkowanych można odwzorować na produkt trzech przestrzeni niemetryzowalnych. Przedstawię rozwiązanie problemu uzyskane wraz z Gonzalo Martinezem. W języku algebr Boole'a wynik daje pewne twierdzenia o zanurzalności produktów wolnych algebr nieprzeliczalnych w produkty algebr interwałowych.

Podczas referatu przypomnimy pojęcie indeksu Szlenka oraz jego związki z asymptotyczną geometrią przestrzeni Banacha. Uzasadnimy, że $c_0$-suma prosta przestrzeni z jednakowo sumowalnym indeksem Szlenka ma sumowalny indeks Szlenka, a także podamy wzór na typ potęgowy ogólnej sumy prostej przestrzeni Banacha

A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of l_1. In this talk I will show some topological properties of this class of compact spaces and its relation with other classes of compact spaces such as Radon-Nikodým or Corson compacta. Most of the results of this talk are contained in the paper 'On weakly Radon-Nikodým spaces' which is available on arxiv.org.