Carlos Uzcategui Aylwin

Industrial University of Santander | UIS · School of Mathematics

We address some phenomena about the interaction between lower semicontinuous submeasures on N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}$$\end{documen...

Let $$\Gamma (X)$$ Γ ( X ) be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space X . There is a topology, denoted $$\tau _\textrm{hco}$$ τ hco , that makes $$\Gamma (X)$$ Γ ( X ) a topological inverse semigroup. We address the question of whether $$\tau _\textrm{hco}$$ τ hco is Polish. For a 0-dimensional...

We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections...

We study the space c0,I of all bounded sequences (xn) that I-converge to 0, endowed with the sup norm, where I is an ideal of subsets of N. We show that two such spaces, c0,I and c0,J , are isometric exactly when the ideals I and J are isomorphic. Additionally, we analyze the connection of the well-known Katětov pre-order ≤K on ideals with some pro...

We study the following reconstruction problem for colorings. Given a countable set X (finite or infinite), a coloring on X is a function $$\varphi : [X]^{2}\rightarrow \{0,1\}$$ φ : [ X ] 2 → { 0 , 1 } , where $$[X]^{2}$$ [ X ] 2 is the collection of all 2-elements subsets of X . A set $$H\subseteq X$$ H ⊆ X is homogeneous for $$\varphi$$ φ when $$...

Let $\Gamma(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $\tau_{hco}$, that makes $\Gamma(X)$ a topological inverse semigroup. We address the question of whether $\tau_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(\Gamma(X),...

We address some phenomena about the interaction between lower semicontinuous submeasures on $\mathbb{N}$ and $F_{\sigma}$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_\sigma$ ideals. We give a partial answers to the question of whether every nonpathological tall $F_\sigma$ ideal is Kat\v{...

The symmetric inverse semigroup I(X) on a set X is the collection of all partial bijections between subsets of X with composition as the algebraic operation. We study the minimal Hausdorff inverse semigroup topology on I(X). We present some characterizations of it. When X is countable such topology is Polish.

We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections...

We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of 𝒮 ⁡ ( G ) on the semigroup I ⁢ ( X ) of partial bijections of X, with 𝒮 ⁡ ( G ) being an inverse monoid introduced by Exel. We show that any unital premorphism θ : G → S , where S is an inver...

We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $\theta:G\to S$, where $S$ is an inv...

Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\b...

The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When $X$ is countable, we show some Polish semigroup topologies on $I(X)$.

Given a hereditarily meager ideal I on a countable set X we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology τI on X such that τI∩I={∅} and, moreover, if I is p+ then τI is selectively separable (SS) and if I is q+, so is τI. In particular, we obtain regular maximal spaces satisfying all boolean combinations of...

Let X be a compact metric countable space, let f:X→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X\rightarrow X$$\end{document} be a homeomorphism and let E(X, f)...

We study a reconstruction problem for colorings. Given a finite or countable set $X$, a coloring on $X$ is a function $\varphi: [X]^{2}\to \{0,1\}$, where $[X]^{2}$ is the collection of all 2-elements subsets of $X$. A set $H\subseteq X$ is homogeneous for $\varphi$ when $\varphi$ is constant on $[H]^2$. Let $hom(\varphi)$ be the collection of all...

We study the concept of a premetric space introduced by F. Richman in the context of constructive mathematics, and present a method for completing them.

We study some variations of the product topology on families of clopen subsets of 2N×N in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle q+.

Given a hereditarily meager ideal $\mathcal{I}$ on a countable set $X$ we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology $\tau^\mathcal{I}$ on $X$ such that $\tau^\mathcal{I}\cap \mathcal{I}=\{\emptyset\}$ and, moreover, if $\mathcal{I}$ is $p^+$ then $\tau^\mathcal{I}$ is selectively separable (SS) and if $\...

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on...

Given a dendrite X and a continuous map f: X → X, we show the following are equivalent: (i) ωf is continuous and Per(f)¯=⋂n∈Nfn(X); (ii) ω(x,f)=Ω(x,f) for each x ∈ X; and (iii) f is equicontinuous. Furthermore, we present some examples illustrating our results.

We study some variations of the product topology on families of clopen subsets of $2^{\mathbb{N}}\times\mathbb{N}$ in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle $q^+$.

An ideal on a set X is a collection of subsets of X closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions. Ideales sobre conjuntos...

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions.

Given a dendrite $X$ and a continuous map $f\colon X\to X$, we show the following are equivalent: (i) $\omega_f$ is continuous and $\overline{\mathrm{Per}(f)}=\bigcap_{n\in\mathbb{N}}f^n(X)$; (ii) $\omega(x,f)=\Omega(x,f)$ for each $x\in X$; and (iii) $f$ is equicontinuous. Furthermore, we present some examples illustrating our results.

Let $X$ be a compact metric countable space and $f:X\to X$ be an homeomorphism. We show that the dynamical system $(X,f)$ is distal if, and only if, every point is periodic. We use this result to give a simpler proof of a theorem of Ellis saying that $(X,f)$ is distal if, and only if, the Ellis semigroup $E(X,f)$ is a group.

We study two form of selective separability, SS and SS⁺, on countable spaces with an analytic topology. We show several Ramsey type properties which imply SS. For analytic spaces X, SS⁺ is equivalent to have that the collection of dense sets is a Gδ subset of 2X, and also equivalent to the existence of a weak base which is an Fσ-subset of 2X. We st...

We present a method for completing a premetric space, in the sense introduced by F. Richman in the context of constructive Mathematics without countable choice.

Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb{N}\}$ are both finite is given. We show that if the collection of all periods...

We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $\tau$ on $X_G$ such that $(X_G, \tau)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,\tau)$. To prove this result we show a generalization of a theorem o...

Proporcionamos condiciones suficientes para que una acción parcial separadamente continua de un grupo de Hausdorff en un espacio métrico sea continua.

We study two form of selective selective separability, $SS$ and $SS^+$, on countable spaces with an analytic topology. We show several Ramsey type properties which imply $SS$. For analytic spaces $X$, $SS^+$ is equivalent to have that the collection of dense sets is a $G_\delta$ subset of $2^X$, and also equivalent to the existence of a weak base w...

Several tasks in artificial intelligence require to be able to find models about knowledge dynamics. They include belief revision, fusion and belief merging, and abduction. In this paper we exploit the algebraic framework of mathematical morphology in the context of propositional logic, and define operations such as dilation or erosion of a set of...

We study possible images of Jones’ set function T . In particular, we are interested in when either T (F1(X)) or T (2X) is finite or countable. We introduce the notion of ω-indecomposable continuum as a generalization of the well known concept of n- indecomposable continuum. We also present results about connectedness and compactness of T (2X). Fin...

Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group G on X. If G and X are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions...

Given a family C of infinite subsets of N, we study when there is a Borel function S:2^N\to 2^N such that for every infinite x\in 2^N, S(x) belongs to C and S(x) is a subset of x. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams' theorem admits such a Borel selector. However, we also...

We show that the following properties are preserved under inverse limits: count-able fan-tightness, q + , discrete generation and selective separability. We also present several examples based on inverse limits of countable spaces.

We present a extension of the classical open mapping principle and Effros' theorem for Polish group actions to the context of partial group actions.

We provide a sufficient condition for a topological partial action of a Polish group on a metric space is continuous, provide that it is separately continuous.

We study several combinatorial properties of (mostly definable) ideals on countable sets. In several cases, we identify critical ideals for such properties in the Katětov order. In particular, the ideal generated by the homogeneous subsets of the random graph is critical for the Ramsey property. The question as to whether there is a tall definable...

We study Borel ideals $I$ on $\mathbb{N}$ with the Fr\'echet property such its orthogonal $I^\perp$ is also Borel (where $A\in I^\perp$ iff $A\cap B$ is finite for all $B\in I$ and $I$ is Fr\'echet if $I=I^{\perp\perp}$). Let $\mathcal{B}$ be the smallest collection of ideals on ${\mathbb{N}}$ containing the ideal of finite sets and closed under co...

Let $X$ be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group $G$ on $X.$ If $G$ and $X$ are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for parti...

Given a continuum $X$, for each $A\subseteq X$, the Jones' set function $\mathcal{T}$ is defined by $\mathcal{T}(A)=\{x\in X : \text{for each subcontinuum }K\text{ such that }x\in \textrm{Int}(K), \text{ then }K\cap A\neq\emptyset\}.$ We show that $\mathcal{D}=\{\mathcal{T}(\{x\}):x\in X\}$ is decomposition of $X$ when $\mathcal{T}$ is continuous....

Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f)^* = E(X,f) \setminus \{f^n : n \in \mathbb{N}\}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a phase space. We show that if $(X,f)$ is a dynamical system such that $X$ is a compact metric countable space and...

A topological space X is said to be maximal if its topology is maximal among all T1T1 topologies over X without isolated points. It is known that a space is maximal if, and only if, it is extremely disconnected, nodec and every open set is irresolvable. We present some results about the complexity of those properties on countable spaces. A countabl...

In this article we attempt to a systematic study of analytic topologies over the natural numbers N (or any countable set X).

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal H is semiselective if and only if Matet's game-theoretic characterization of the H-Ramsey property holds. This lifts...

We present results about the Cantor-Bendixson index of some subspaces of a uniform family F of finite subsets of natural numbers with respect to the lexicographic order topology. As a corollary of our results we get that for any omega-uniform family F the restriction F|M is homeomorphic to F iff M contains intervals of arbitrary length of consecuti...

It is shown that Matet's characterization of $\mathcal{H}$-Ramseyness relative to a selective coideal $\mathcal{H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a coideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic characterization of $\mathcal{H}...

A space is called subsequential if it is a subspace of a sequential space. A free filter F on ω is called subsequential if the space ω∪{F} is subsequential. The purpose of this paper is to introduce the degree of subsequentiality of a subsequential filter in a similar way as it is done in the realm of sequential spaces. A method to produce subseque...

Following N. Noble, we say that a space is subsequential if it is a subspace of a sequential space. A free filter F on ω is called subsequential if the space ω∪{F} is subsequential. In this paper, we state several properties of these filters.

On the complexity of the family of compact subsets of Q Raúl Naulin and Carlos Uzcátegui Aylwin Resumen Mostramos que K(Q), la familia de subconjuntos compactos de Q, es Π 1 1-completa en el cubo de Cantor 2 Q. Palabras claves: Conjuntos coanalíticos completos, Teorema de Hurewicz, Teoría descriptiva de conjuntos. Abstract We show that K(Q), the co...

We show several representation theorems for explanatory reasoning based on cumulative models. An explanatory process is given by a binary relation ▷ between formulas in a propositional language where the intended meaning of α▷γ is “γ is a preferred explanation of α”. To each cumulative model E (a variation of those studied by Kraus, Lehmann and Mag...

Let Dk(w)Dk(w) be the multiset containing all factors of ww of length kk including repetitions. One of the main results is that if Dk(w)=Dk(v)Dk(w)=Dk(v) for all k≤⌊|w|2⌋+1, then w=vw=v. The bound ⌊|w|2⌋+1 is optimal; however we will also show that if Dk(w)=Dk(v)Dk(w)=Dk(v) for all k≤⌊|w|2⌋, then ww and vv are structurally similar.

In the context of a generalized topology g on a set X, we give in this article characterizations of some separation axioms between T0 and T2 in terms of properties of the diagonal in X × X.

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In this paper we introduce a median operator between two sets of interpretations (worlds) in a finite propositional language. Our definition is based on morphological operations and Hausdorff distance. It provides a result which lies “halfway” between both sets and accounts for the “extension” or “shape” of the sets. We prove several interesting pr...

A topological space X is said to be generated by an ideal I if for all A ⊆ X and all x ∈ A there is E ⊆ A in I such that x ∈ E, and is said to be weakly generated by I if whenever a subset A of X contains E for every E ⊆ A with E ∈ I, then A itself is closed. An important class of examples are the so called weakly discretely generated spaces (which...

We study the behavior of the Randić index χ subject to the operation on a tree T which creates a new tree T′≠T by deleting an edge ax of T and adding a new edge incident to either a or x. Let ≼mso be the smallest poset containing all pairs (T,T′) such that χ(T)<χ(T′) and T,T′∈Cn (where Cn is the collection of trees with n vertices and of maximum de...

We study sequential convergence in spaces with analytic topologies avoiding thus a number of standard pathologies. For example, we identify bisequentiality of an analytic space as the Fréchet property of its square. We show that a countable Fréchet group is metrizable if and only if its topology is analytic. We also investigate the diagonal sequenc...

We present some results about the class of Alexandroff topologies (i.e. topologies where the intersection of arbitrary many open sets is open) from the perspective obtained when they are viewed as closed subsets of the Cantor cuber 2X (the power set of X with the product topology).

We present a generalization of the following result of Y. Benyamini. There is a continuous function $f:\R\rightarrow \R$ such that for each $(x_n)_{n\in\Z}\in [0,1]^\Z$, there is $t\in\R$ such that $x_n=f(t+n)$ for all $n\in \Z$.

Abstract Using morpho-logics we show how to find explana- tions of observations, how to perform revision, con- traction, fusion, in an unified way. In the framework of abduction, we show how to deal with observa- tions inconsistent with the background,theory and introduce methods,to treat multiple observations. Based on these ideas we introduce a d...

In abductive reasoning, preference criteria for selecting the best explanation are regarded as qualitative properties (like being simpler or more plausible) which are beyond the pure causal or deductive relationship between an explanandum and its explanations. This paper is a contribution to the clarification of the relationship between preference...

Let G be a collection of graphs with n vertices. We present a simple description of [G]χ={H∈G:χ(H)=χ(G)} where χ denotes the Randić index. We associate to G a Q-linear map ρ:Qm→Qk (for some integers k,m depending on G) such that the kernel of ρ contains the necessary information to describe [G]χ in terms of linear equations. These results provide p...

Let (X, τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set 2X (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel'skiǐ-Franklin space Sω is Fσδ. In this paper we study the complexity, in the sense of the Borel hierarchy,...

Two topologies τ and ρ over X are said to be complementary if τ ∧ρ is the indiscrete topology and τ ∨ρ the discrete topology. The lattice of topologies is complemented, i.e, every topology has a complement. We will show that every AT topology (i.e. a topology such that the intersection of arbitrary many open sets is open) over a countable set has a...

Using mathematical morphology on formulas introduced recently by Bloch and Lang (Proceedings of IPMU’2000) we define two new explanatory relations. Their logical behavior is analyzed. The results show that these natural ways for defining preferred explanations are robust because these relations satisfy almost all postulates of explanatory reasoning...

In this article we attempt to a systematic study of analytic topologies over the natural numbers (or any countable set X).

One of the main tools in the study of nonmonotonic consequence relations is the representation of such relations in terms of preferential models. In this paper we give an unified and simpler framework to obtain such representation theorems.

One of the main tools in the study of non-monotonic consequence relations is the representation of such relations in terms of preferential models. In this paper we give an unified and simpler framework to obtain such representation theorems.

We study the relationship between some structural rules for abduc-tive reasoning and preference relations for selection preferred explanations. We prove that explanatory relations having good structural properties can always be defined by orders over formulas.

Abduction is usually defined as the process of inferring the best explanation of an observation. There are many information processing operations that can be viewed as a search for an explanation. For instance, diagnosis, natural language interpretation and plan recognition. This paper is concerned about the following aspects of abduction: (i) what...

In this paper we present a systematic study of abductive consequence relations. We show that a monotone abductive consequence relation satisfies the properties of a cumulative monotonic system as defined by Kraus, Lehmann and Magidor when the disjunction of all abductive explanations is the explanation used to justify the observations. We also show...

A collection of topologies Φ α (for α an ordinal) is introduced in the space of bounded continuous functions C b (X) (where X is a discrete space). It is proved that |X| ≤ ℵ α if and only if the unit ball B 1 (X) in C b (X) is Φ α-compact. We compute the dual of (C b (X), Φ α) and present a characterization of the cofinality of |X| in terms of Φ 0-...

This paper describes a change theory based on abductive reasoning. We take the AGMpostulates for revisions, expansions and contractions, and Katsuno and Mendelzon postulatesfor updates and incorporate abduction into them. A key feature of the theory is that presentsa unified view of standard change operators and abductive change operators rather th...

We study some properties of smooth Borel sets with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets. We generalize a theorem of Weiss from Z-actions t...

We study normal filters on the set spaces λ , P κ ( λ ) , [ λ ] κ \lambda ,{\mathcal {P}_\kappa }\left ( \lambda \right ),{\left [ \lambda \right ]^\kappa } , and ( λ ) κ {\left ( \lambda \right )^\kappa } . We characterize the least normal γ \gamma -complete filter containing a given γ \gamma -complete filter for γ ≥ ω 1 \gamma \geq {\omega _1} ....

We study normal filters on the set spaces λ, Pκ(λ), [ λ ]κ, and (λ)κ. We characterize the least normal γ-complete filter containing a given γ-complete filter for γ ≥ ω1. If F is a ω1-complete filter on any of the set spaces mentioned, the least ω1-complete normal filter containing it is the filter generated by the sets $\{x \in E\mid\alpha_1,\ldots...

We study the relationship between some structural rules for abduc- tive reasoning and preference relations for selection preferred explana- tions. We prove that explanatory relations having good structural properties can always be defined by orders over formulas.

Thesis (Doctor of Philosophy)-- California Institute of Technology, California, 1990 Incluye bibliografía

Mecanografiado Trabajo de ascenso (Prof. Asociado)-- Universidad de Los Andes, Facultad de Ciencias. Departamento de Matemáticas, Mérida, 1995 Incluye bibliografía

Mecanografiado Tesis (Lic. en Matemáticas)-- Universidad de Los Andes, Facultad de Ciencias, Departamento de Matemáticas, Grupo de Análisis Funcional, Mérida, 2004 Incluye bibliografía

Tesis (Magister Scientiarum)-- Instituto Venezolano de Investigaciones Científicas, Caracas, 1985 Incluye bibliografía

Cuarta escuela venezolana de Matematicos, Mérida 4 al 14 de Septiembre de 1991 Incluye bibliografía

Incluye bibliografía

Mecanografiado Tesis (Lic. en Matemáticas) -- Universidad de Los Andes, Facultad de Ciencias, Departamento de Matemáticas, Mérida, 2005 Incluye bibliografía