Search

A two-point selection on a set $X$ is a function $f:[X]^2 \to X$ such that $f(F) \in F$ for every $F \in [X]^2$. It is known that every two-point selection $f:[X]^2 \to X$ induced a topology $\tau_f$ on $X$ by using the relation: $x \leq y$ if either $f(\{x,y\}) = x$ or $x = y$, for every $x, y \in X$. We are mainly concern with the two-point selections on the real line $\mathbb{R}$. In this paper, we study the $\sigma$-algebras of Borel, each one denoted by $\mathcal{B}_f(\mathbb{R})$, of the topologies $\tau_f$'s defined by a two-point selection $f$ on $\mathbb{R}$. We prove that the assumption $\mathfrak{c} = 2^{< \mathfrak{c}}$ implies the existence of a family $\{ f_\nu : \nu < 2^\mathfrak{c} \}$ of two-point selections on $\mathbb{R}$ such that $\mathcal{B}_{f_\mu}(\mathbb{R}) \neq \mathcal{B}_{f_\nu}(\mathbb{R})$ for distinct $\mu, \nu < 2^\mathfrak{c}$. By assuming that $\mathfrak{c} = 2^{< \mathfrak{c}}$ and $\mathfrak{c}$ is regular, we also show that there are $2^{2^\mathfrak{c}}$ many $\sigma$-algebras on $\mathbb{R}$ that contain $[\mathbb{R}]^{\leq \omega}$ and none of them is the $\sigma$-algebra of Borel of $\tau_f$ for any two-point selection $f: [\mathbb{R}]^2 \to \mathbb{R}$. Several examples are given to illustrate some properties of these Borel $\sigma$-algebras.

A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of $(k,\varepsilon)-$oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where $(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using what we will call blocks of barriers. In particular, we prove that the Ramsey Theorem is equivalent to the statement ``for every finite sequence $(\mathcal{B}_i)_{i=1}^k$ of barriers, every $\varepsilon>0$ and every normalized sequence $(x_i)_{i\in\mathbb{N}}$ there is a subsequence $(x_i)_{i\in M}$ that is $((\mathcal{B}_i\cap\mathcal{P}(M))_{i=1}^k,\varepsilon)-$block oscillation stable'', where $\mathcal{P}(M)$ is the power set of the infinite set M. Besides, we introduce the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic model of a normalized basic sequence where $(\mathcal{B}_i)_{i\in\mathbb{N}}$ is a sequence of barriers. These models are a generalization of the spreading models and are related to the $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequences. We show that the Brunel-Sucheston is satisfied for the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models, and we also prove that this result is equivalent to the Ramsey Theorem. The difference between our theorem and the Brunel-Sucheston Theorem is based on the number of different models that are obtained from the same normalized basic sequence through them. This and other observations about $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models are noted in an example at the end of the article.

It is know that the Valdivia compact spaces can be characterized by a special family of retractions called $r$-skeleton (see \cite{kubis1}). Also we know that there are compact spaces with $r$-skeletons which are not Valdivia. In this paper, we shall study $r$-squeletons and special families of closed subsets of compact spaces. We prove that if $X$ is a zero-dimensional compact space and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$ such that $|r_s(X)| \leq \omega$ for all $s\in \Gamma$, then $X$ has a dense subset consisting of isolated points. Also we give conditions to an $r$-skeleton in order that this $r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a $\Sigma$-product of a power of the unit interval. Following this fact we introduce the notion of $\pi$-skeleton on a compact space $X$ by embedding $X$ in a suitable power of the unit interval together with a pair $(\mathcal{F},\varphi)$, where $\mathcal{F}$ is family of metric separable subspaces of $X$ and $\varphi$ an $\omega$-monotone function which satisfy certain properties. This new notion generalize the idea of a $\Sigma$-product. We prove that a compact space admits a retractional-skeleton iff it admits a $\pi$-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with retractional-skeletons admits an retractional-skeleton (see \cite{cuth1}). In \cite{casa1}, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak $c$-skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with $r$-skeletons., Comment: arXiv admin note: text overlap with arXiv:1804.01549

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 a point $a\in X'$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical systems. We show that if $(X,f)$ is a transitive dynamical system and either every function of $E(X,f)$ is continuous or $|\omega_f(x)|=1$ for each accumulation point $x$ of $X$, then $E(X,f)$ is homeomorphic to $X$. Several examples are given to illustrate our results., Comment: 12 pages

It is known that there exists a Banach space $X$ with a Schauder basis $(e_i)_{i=1}^{\infty}$ which does not admit $\ell_p$ as the model space obtained by a finite chain of sequences such that each element is a spreading model of a block subsequence of the previous element, starting from a block subsequence of $(e_i)_{i=1}^{\infty}$. We prove that $X$ has the stronger property of not admitting $\ell_p$ via a finite chain consisting of block asymptotic models. This is related to a question posed by L. Halbeisen and E. Odell for the special case of block generated asymptotic models. Also, we show that for every $k \in \mathbb{N}$ the Ramsey Coloring Theorem for $[\mathbb{N}]^{k}$ is equivalent to the following $k$-oscillation stability: In an arbitrary Banach space $X$, for every $\epsilon > 0$ and for every normalized sequence $(e_i)_{i \in \mathbb{N}}$ in $X$ there exists $M \in [\mathbb{N}]^{\infty}$ such that if $n_1,n_2,\cdots,n_k,m_1,m_2,\cdots,m_k \in M$, then $$ \big| ||\sum_{i =1}^{k}a_i e_{n_i} || - ||\sum_{i=1}^{k} a_i e_{m_i} || \big| < \epsilon $$ for all $(a_i)_{i=1}^{k} \in [-1,1]^k$.

An $r$-skeleton on a compact space is a family of continuous retractions having certain rich properties. The $r$-skeletons have been used to characterized the Valdivia compact spaces and the Corson compact spaces. Here, we characterized a compact space with an $r$-skeleton, for which the given $r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of the given space. Besides, we prove that if $X$ is a zero-dimensional compact space without isolated points and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$, then there is $s\in \Gamma$ such that $cl(r_s[X])$ is not countable.

A space $X$ is called {\it selectively pseudocompact} if for each sequence $(U_{n})_{n\in \mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\in \mathbb{N}}$ of points in $X$ such that $cl_X(\{x_n : n < \omega\}) \setminus \big(\bigcup_{n < \omega}U_n \big) \neq \emptyset$ and $x_{n}\in U_{n}$, for each $n < \omega$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power is countably compact but its $(k+1)$-st power is not selectively pseudocompact. This provides a positive answer to a question posed in \cite{gt} in any model of $ZFC+CH$.

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 of the periodic points of $(X,f)$ is infinite, then $E(X,f)$ has size $2^{\aleph_0}$. It is also proved that if $(X,f)$ has a point with a dense orbit and all elements of $E(X,f)$ are continuous, then $|E(X,f)| \leq |X|$. For dynamical systems of the form $(\omega^2 +1,f)$, we show that if there is a point with a dense orbit, then all elements of $E(\omega^2+1,f)$ are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where $E(\omega^2+1,f)$ and $\omega^2+1$ are homeomorphic but not algebraically homeomorphic, where $\omega^2+1$ is taken with the usual ordinal addition as semigroup operation., Comment: 15 pages

In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a Fre\'ech-Urysohn space $X$, this hyperspace is equipped with the Vietoris topology. We mainly prove that $\mathcal{S}_c(X)$ is meager whenever $X$ is a crowded space, as a corollary we obtain that if $\mathcal{S}_c(X)$ is Baire, the $X$ has a dense subset of isolated points. As an interesting example $\mathcal{S}_c(\omega_1)$ has the Baire property, where $\omega_1$ carries the order topology (this answers a question from \cite{sal-yas}). We can give more examples like this one by proving that the Alexandroff duplicated $\mathcal{A}(Z)$ of a space $Z$ satisfies that $\mathcal{S}_c(\mathcal{A}(Z))$ has the Baire property, whenever $Z$ is a $\Sigma$-product of completely metrizable spaces and $Z$ is crowded. Also we show that if $\mathcal{S}_c(X)$ is pseudocompact, then $X$ has a relatively countably compact dense subset of isolated points, every finite power of $X$ is pseudocompact, and every $G_\delta$-point in $X$ must be isolated. We also establish several topological properties of the hyperspace of nontrivial convergent sequences of countable Fre\'ech-Urysohn spaces with only one non-isolated point.

A {\it weak selection} on $\mathbb{R}$ is a function $f: [\mathbb{R}]^2 \to \mathbb{R}$ such that $f(\{x,y\}) \in \{x,y\}$ for each $\{x,y\} \in [\mathbb{R}]^2$. In this article, we continue with the study (which was initiated in \cite{ag}) of the outer measures $\lambda_f$ on the real line $\mathbb{R}$ defined by weak selections $f$. One of the main results is to show that $CH$ is equivalent to the existence of a weak selection $f$ for which: \[ \mathcal \lambda_f(A)= \begin{cases} 0 & \text{if $|A| \leq \omega$,}\\ \infty & \text{otherwise.} \end{cases} \] Some conditions are given for a $\sigma$-ideal of $\mathbb{R}$ in order to be exactly the family $\mathcal{N}_f$ of $\lambda_f$-null subsets for some weak selection $f$. It is shown that there are $2^\mathfrak{c}$ pairwise distinct ideals on $\mathbb{R}$ of the form $\mathcal{N}_f$, where $f$ is a weak selection. Also we prove that Martin Axiom implies the existence of a weak selection $f$ such that $\mathcal{N}_f$ is exactly the $\sigma$-ideal of meager subsets of $\mathbb{R}$. Finally, we shall study pairs of weak selections which are "almost equal" but they have different families of $\lambda_f$-measurable sets.

We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from \cite{sal-rey}. Besides, we introduce the notion of full $c$-skeleton and prove that a compact space is Corson if and only if it has a full $c$-skeleton.

We introduce the {\it Ellis semigroup} of a nonautonomous discrete dynamical system $(X,f_{1,\infty})$ when $X$ is a metric compact space. The underlying set of this semigroup is the pointwise closure of $\{f\sp{n}_1 \, |\, n\in \mathbb{N}\}$ in the space $X\sp{X}$. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system.

A filter $\F$ on $\w$ is called Fr\'echet-Urysohn if the space with only one non-isolated point $\w \cup \{\F\}$ is a Fr\'echet-Urysohn space, where the neighborhoods of the non-isolated point are determined by the elements of $\F$. In this paper, we distinguish some Fr\'echet-Urysohn filters by using two pre-orderings of filters: One is the Rudin-Keisler pre-order and the other one was introduced by Todor\v{c}evi\'c-Uzc\'ategui in \cite{tu05}. In this paper, we construct an $RK$-chain of size $\c^+$ which is $RK$-above of avery $FU$-filter. Also, we show that there is an infinite $RK$-antichain of $FU$-filters.

This paper is a continuation of the work done in \cite{sal-yas} and \cite{may-pat-rob}. We deal with the Vietoris hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a space $X$. We answer some questions in \cite{sal-yas} and generalize several results in \cite{may-pat-rob}. We prove that: The connectedness of $X$ implies the connectedness of $\mathcal{S}_c(X)$; the local connectedness of $X$ is equivalent to the local connectedness of $\mathcal{S}_c(X)$; and the path-wise connectedness of $\mathcal{S}_c(X)$ implies the path-wise connectedness of $X$. We also show that the space of nontrivial convergent sequences on the Warsaw circle has $\mathfrak{c}$-many path-wise connected components, and provide a dendroid with the same property.

In this article, we mainly study certain families of continuous retractions ($r$-skeletons) having certain rich properties. By using monotonically retractable spaces we solve a question posed by R. Z. Buzyakova in \cite{buz} concerning the Alexandroff duplicate of a space. Certainly, it is shown that if the space $X$ has a full $r$-skeleton, then its Alexandroff duplicate also has a full $r$-skeleton and, in a very similar way, it is proved that the Alexandroff duplicate of a monotonically retractable space is monotonically retractable. The notion of $q$-skeleton is introduced and it is shown that every compact subspace of $C_p(X)$ is Corson when $X$ has a full $q$-skeleton. The notion of strong $r$-skeleton is also introduced to answer a question suggested by F. Casarrubias-Segura and R. Rojas-Hern\'andez in their paper \cite{cas-rjs} by establishing that a space $X$ is monotonically Sokolov iff it is monotonically $\omega$-monolithic and has a strong $r$-skeleton. The techniques used here allow us to give a topological proof of a result of I. Bandlow \cite{ban} who used elementary submodels and uniform spaces.

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 every accumulation point $X'$ is periodic, then either each function of $E(X,f)^*$ is continuous or each function of $E(X,f)^*$ is discontinuous. We describe an example of a dynamical system $(X,f)$ where $X$ is a compact metric countable space, the orbit of each accumulation point is finite and $E(X,f)^*$ contains continuous and discontinuous functions.