arXiv subjects
Publications and source records attributed to Wiesław Kubiś.
We discuss some finite homogeneous structures, addressing the question of universality of their automorphism groups. We also study the existence of so-called Katětov functors in finite categories of embeddings or homomorphisms.
We study the abstract Banach-Mazur game played with finitely generated structures instead of open sets. We characterize the existence of winning strategies aiming at a single countably generated structure. We also introduce the concept of weak Fraisse classes, extending the classical Fraisse theory and revealing its relations to our Banach-Mazur game.
We study a metric-like structure on categories, showing that the concept of the limit of a sequence in a metric space and the concept of the colimit of a sequence in a category have a common generalization. The main concept is a norm on a category, generalizing pseudo-metrics and group valuations. In this new context, we discuss topics like Cauchy completion and the Banach Contraction Principle.
We describe the Lelek fan, a smooth fan whose set of end-points is dense, and the Poulsen simplex, a Choquet simplex whose set of extreme points is dense, as Fraïssé limits in certain natural categories of embeddings and projections. As an application we give a short proof of their uniqueness, universality, and almost homogeneity. We further show that for every two countable dense subsets of end-points of the Lelek fan there exists an auto-homeomorphism of the fan mapping one set onto the other. This improves a result of Kawamura, Oversteegen, and Tymchatyn from 1996.
Let $U_{FNA}$ be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurarii Banach space $G$ over a non-archimedean valued field $K$ is constructed, i.e. a non-archimedean Banach space $G$ of countable type which is of almost universal disposition for the class $U_{FNA}$. This means: for every isometry $g : X \to Y$ , where $Y \in U_{FNA}$ and $X$ is a subspace of $G$, and every $\varepsilon \in (0,1)$ there exists an $\varepsilon$-isometry $f : Y \to G$ such that $f(g(x)) = x$ for all $x \in X$. We show that all non-archimedean Banach spaces of almost universal disposition for the class $U_{FNA}$ are $\varepsilon$-isometric. Furthermore, all non-archimedean Banach spaces of almost universal disposition for the class $U_{FNA}$ are isometrically isomorphic if and only if $K$ is spherically complete and $\{|λ| : λ\in K \setminus \{0\} \} = (0,\infty)$.
We prove a result on perfect cliques with respect to countably many G-delta relations on a complete metric space. As an application, we show that a Polish group contains a free subgroup generated by a perfect set as long as it contains any uncountable free subgroup. This answers a recent question of Głcab and Strobin.
We develop a theory of \emph{Katetov functors} which provide a uniform way of constructing Fraisse limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Fraisse limits.
We show that for each $p\in(0,1]$ there exists a separable $p$-Banach space $\mathbb G_p$ of almost universal disposition, that is, having the following extension property: for each $ε>0$ and each isometric embedding $g:X\to Y$, where $Y$ is a finite dimensional $p$-Banach space and $X$ is a subspace of $\mathbb G_p$, there is an $ε$-isometry $f:Y\to \mathbb G_p$ such that $x=f(g(x))$ for all $x\in X$. Such a space is unique, up to isometries, does contain an isometric copy of each separable $p$-Banach space and has the remarkable property of being "locally injective" amongst $p$-Banach spaces. We also present a nonseparable generalization which is of universal disposition for separable spaces and "separably injective". No separably injective $p$-Banach space was previously known for $p<1$.
Given a compact space in a fixed universe of set theory, one can naturally define its interpretation in any ZFC extension of the universe. We investigate the stability of some classes of compact spaces with respect to extensions of this sort. We show that the class of Eberlein/Gul'ko compacta is stable (= absolute). On the other hand, there are examples of Corson compacta which are no longer Corson in some forcing extensions. All the material comes from the author's notes written between 2004 -- 2006.
We study an analogue of the Parovicenko property in categories of compact spaces with additional structures. In particular, we present an internal characterization of this property in the class of compact median spaces.
We construct a nonexpansive linear operator on the Gurarii space that "captures" all nonexpansive linear operators between separable Banach spaces. Some additional properties involving its restrictions to finite-dimensional subspaces describe this operator uniquely up to an isometry.
We develop the theory of approximate Fraïssé limits in the context of categories enriched over metric spaces. Among applications, we construct a generic projection on the Gurarii space and we present a simpler proof of a characterization of the pseudo-arc, due to Irwin and Solecki.
Some new classes of compacta $K$ are considered for which $C(K)$ endowed with the pointwise topology has a countable cover by sets of small local norm--diameter.
We study dualities between classes of relational topological structures, given by Hom-functors. We show that there exists a 2-element structure with infinitely many relations, which reconstructs all other structures generated by a 2-element one. As an application, we find a natural duality for the class of normal convexity structures. As another application, we give short proofs for several known dualities for classes of structures generated by a fixed 2-element structure.
We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindelöf in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality $ω_1$ which induce monolithic spaces of the form $K_{\mathcal A}$: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations.
We present selected known results and some of their improvements, involving Gurarii spaces. A Banach space is Gurarii if it has certain natural extension property for almost isometric embeddings of finite-dimensional spaces. Deleting the word "almost", we get the notion of a strong Gurarii space. There exists a unique (up to isometry) separable Gurarii space, however strong Gurarii spaces cannot be separable. The structure of the class of non-separable Gurarii spaces seems to be not very well understood. We discuss some of their properties and state some open questions. In particular, we characterize non-separable Gurarii spaces in terms of skeletons of separable subspaces, we construct a non-separable Gurarii space with a projectional resolution of the identity and we show that no strong Gurarii space can be weakly Lindelöf determined.
We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.
We study some topological properties of trees with the interval topology. In particular, we characterize trees which admit a 2-fibered compactification and we present two examples of trees whose one-point compactifications are Rosenthal compact with certain renorming properties of their spaces of continuous functions.