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We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. We show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness. Further, we introduce and study the notions of chain union stability and of chain union rank, which measure how often the process of closing a ball space under all unions of chains has to be iterated until a ball space is obtained that is closed under unions of chains., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1810.05832

We show that an embedding of a fixed 0-dimensional compact space $K$ into the \v{C}ech--Stone remainder $\omega^*$ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from $K$ to compact metric spaces. Using Fra\"iss\'e theory we get a few well know theorems about \v{C}ech--Stone remainder. We establish the following: -- an ultrametric space $K$ of weight $\kappa$ can be uniformly embedded into $\kappa^\kappa$ as a uniformly nowhere dense subset, -- every uniform homeomorphism of uniformly nowhere dense sets in $\kappa^\kappa$ can be extended to a uniform auto-homeomorphism of $\kappa^\kappa$, -- every uniformly nowhere dense set in $\kappa^\kappa$ is a uniform retract of $\kappa^\kappa$. If we assume that $\kappa$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption.

Motivated by a recent work of Balcerzak and Kania [Proc. Amer. Math. Soc. 151 (2023) 3737--3742], we show that every countable monoid has a universal action on the free object over a countable infinite set. This is a general result concerning concrete categories with a left adjoint (free) functor. On the way, we introduce an abstract concept of ``being generated by a set". At the same time we obtain a simpler proof of the result of Darji and Matheron concerning a surjectively universal operator on the classical Banach space $\ell_1$ of summable sequences., Comment: 13 pages

We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to isometry as vector spaces over the two-element field, endowed with an injective norm. Using isosceles-free decompositions, we provide bounds on the maximal number of distances in arbitrary homogeneous finite metric spaces., Comment: 36 pages

Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We also study the corresponding profinite graphs. Finally, we present a concrete model of the projectively universal profinite graph, the projective Fraisse limit of finite graphs, showing in particular that it has a dense subset of isolated vertices., Comment: 35 pages

We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set of primes P) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player wins if the limit of this sequence is homeomorphic to a concrete (fixed in advance) space, called generic whenever the second player has a winning strategy. For this aim, we develop a new approximate Fra\"iss\'e theory, in order to realize the above-mentioned objects (the pseudo-arc and the pseudo-solenoids) as Fra\"iss\'e limits. Our framework extends the discrete Fra\"iss\'e theory, both classical and projective, and is also suitable for working directly with continuous maps on metrizable compacta. We show, in particular, that, when playing with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic. The universal pseudo-solenoid appears to be generic over all surjections between circle-like continua., Comment: 64 pages, 2 figures, comments are welcome

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that every finite linear space embeds into a finite projective plane., Comment: 21 pages

We introduce natural strengthenings of sequential compactness called the $r$-Ramsey property for each natural number $r\geq 1$. We prove that metrizable compact spaces are $r$-Ramsey for all $r$ and give examples of compact spaces that are $r$-Ramsey but not $r+1$-Ramsey for each $r\geq 1$ (assuming CH for all $r>1$, Comment: 10 pages

We extend the Kechris--Pestov--Todor\v{c}evi\'c correspondence to weak Fra\"{\i}ss\'{e} categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders, and categories of strong embeddings of trees., Comment: 57 pages, 12 figures

We introduce the concept of an abstract evolution system, which provides a convenient framework for studying generic mathematical structures and their properties. Roughly speaking, an evolution system is a category endowed with a selected class of morphisms called transitions, and with a selected object called the origin. We illustrate it by a series of examples from several areas of mathematics. We formulate sufficient conditions for the existence of the unique "most complicated" evolution. In case the evolution system "lives" in model theory and nontrivial transitions are one-point extensions, the limit of the most complicated evolution is known under the name Fraisse limit, a unique countable universal homogeneous model determined by a fixed class of finitely generated models satisfying some obvious axioms. Evolution systems can also be viewed as a generalization of abstract rewriting systems, where the partially ordered set is replaced by a category. In our setting, the process of rewriting plays a nontrivial role, whereas in rewriting systems only the result of a rewriting procedure is relevant. An analogue of Newman's Lemma holds in our setting, although the proof is a bit more delicate, nevertheless, still based on Huet's idea using well founded induction., Comment: New examples added; different title; more topics in the last section; 38 pages

We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindel\"of number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection properties., Comment: Some corrections and clarifications, 33 pages

Inspired by classic work of Wallman and more recent work of Jung-Kegelmann-Moshier and Vickers, we show how to encode general subbases of stably locally compact spaces via certain entailment relations. We further build this up to a categorical duality encompassing the classic Priestley-Stone duality and its various extensions to stably locally compact spaces by Shirota, De Vries, Hofmann-Lawson (in the stable case), Jung-S\"underhauf, Hansoul-Poussart, Bezhanishvili-Jansana, van Gool and Bice-Starling.

We study betweenness preserving mappings (we call them \emph{monotone}) defined on subsets of the plane. Once the domain is a convex set, such a mapping is either the restriction of a homography, or its image is contained in the union of a line and a single point, or its image consists of five points, one of them being between two disjoint pairs of the other four points. We also show that an open planar set cannot be mapped in a one-to-one monotone way into the real line. From this we deduce that a one-to-one monotone mapping from a convex planar set with nonempty interior is necessarily a partial homography. Finally, we prove that a set consisting of three pairwise non-parallel lines does not admit a one-to-one monotone mapping into the real line, while on the other hand a set consisting of three closed line segments intersecting at a single point does admit such a mapping., Comment: Final version (21 pages, 4 figures)

We discuss some finite homogeneous structures, addressing the question of universality of their automorphism groups. We also study the existence of so-called Kat\v{e}tov functors in finite categories of embeddings or homomorphisms., Comment: 9 pages

We extend Wallman's classic duality from lattice bases to semilattice subbases and from compact to locally closed compact spaces. Moreover, we make this duality functorial via appropriate relational morphisms.

We compare two types of universal operators constructed relatively recently by Cabello S\'anchez, and the authors. The first operator $\Omega$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $\Omega$ is generic in the sense of a natural infinite game., Comment: 12 pages

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit., Comment: Some additions and tiny corrections; 15 pages

We study the approximately finite-dimensional (AF) $C^*$-algebras that appear as inductive limits of sequences of finite-dimensional $C^*$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra $\mathcal A_\mathfrak{F}$ with the property that any separable AF-algebra is isomorphic to a quotient of $\mathcal A_\mathfrak{F}$. Equivalently, by Elliott's classification of separable AF-algebras, there are surjectively universal countable scaled (or with order-unit) dimension groups. This universality is a consequence of our result stating that $\mathcal A_\mathfrak{F}$ is the Fra\"\i ss\'e limit of the category of all finite-dimensional $C^*$-algebras and left-invertible embeddings. With the help of Fra\"\i ss\'e theory we describe the Bratteli diagram of $\mathcal A_\mathfrak{F}$ and provide conditions characterizing it up to isomorphisms. $\mathcal A_\mathfrak{F}$ belongs to a class of separable AF-algebras which are all Fra\"\i ss\'e limits of suitable categories of finite-dimensional $C^*$-algebras, and resemble $C(2^\mathbb N)$ in many senses. For instance, they have no minimal projections, tensorially absorb $C(2^\mathbb N)$ (i.e. they are $C(2^\mathbb N)$-stable) and satisfy similar homogeneity and universality properties as the Cantor set., Comment: The content is the same as in the published version

We study spherical completeness of ball spaces and its stability under expansions. We introduce the notion of an ultra-diameter, mimicking diameters in ultrametric spaces. We prove some positive results on preservation of spherical completeness involving ultra-diameters with values in narrow partially ordered sets. Finally, we show that in general, chain intersection closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness., Comment: 11 pages

We develop the theory of weak Fraisse categories, where the crucial concept is the weak amalgamation property, discovered relatively recently in model theory. We show that, in a suitable framework, every weak Fraisse category has its unique limit, a special object in a bigger category, characterized by certain variant of injectivity. This significantly extends the present theory of Fraisse limits., Comment: Several examples and comments added; new subsection 6.2 (38 pages)

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., Comment: 20 pages

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., Comment: Last section extended (16 pages)

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 $\{|\lambda| : \lambda \in K \setminus \{0\} \} = (0,\infty)$., Comment: 13 pages

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 Fra\"{i}ss\'e 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., Comment: version accepted to RACSAM

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 G{\l}\c{a}b and Strobin., Comment: 10 pages

We present a version of the Banach-Mazur game, where open sets are replaced by elements of a fixed partially ordered set. We show how to apply it in the theory of Fraisse limits and beyond, obtaining simple proofs of universality of certain objects and classes., Comment: 13 pages

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., Comment: Complete version, added more material, small corrections; 40 pages, 1 figure

In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches., Comment: 20 pages

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., Comment: 15 pages

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., Comment: 17 pages, extended subsection 2.2. Final version

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 $\epsilon>0$ 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 $\epsilon$-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 $p<1$., Comment: Final version. Minor corrections, the proof of Corollary 6.6 expanded (24 pages)

We prove that a separable Hausdorff topological space $X$ containing a cocountable subset homeomorphic to $[0,\omega_1]$ admits no separately continuous mean operation and no diagonally continuous $n$-mean for $n\ge 2$., Comment: 6 pages

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., Comment: Minor revision, final version, 19 pages

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., Comment: 11 pages

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., Comment: 24 pages

We develop the theory of approximate Fra\"{i}ss\'{e} 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., Comment: Corrected abstract and some typos

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 Lindel\"of 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 $\omega_1$ 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., Comment: 21 pages

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 Lindel\"of determined., Comment: 30 pages

We present a short and elementary proof of isometric uniqueness of the Gurarii space., Comment: 6 pages, some improvements incorporated

We present a purely category-theoretic characterization of retracts of Fra\"iss\'e limits. For this aim, we consider a natural version of injectivity with respect to a pair of categories (a category and its subcategory). It turns out that retracts of Fra\"iss\'e limits are precisely the objects that are injective relatively to such a pair. One of the applications is a characterization of non-expansive retracts of Urysohn's universal metric space., Comment: 33 pages; minor corrections; extended introduction and more references. Final version

We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton., Comment: 25 pages, One reference added

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., Comment: 12 pages

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., Comment: Small changes, mainly in the introduction and in final remarks