arXiv subjects

Alejandro Ríos-Herrejón

Publications and source records attributed to Alejandro Ríos-Herrejón.

10 recordsLinked to original sources

arXiv2026

Super calibers in topological spaces and topological hyperspaces

We study the notion of a super caliber of a topological space, which is closely related to the classical notion of caliber and has appeared in the literature under several different names. We collect and unify several known results and establish new results concerning the collections of super calibers of topological spaces and their hyperspaces. In particular, we investigate the relationship between the super calibers of a space $X$ and those of hyperspaces $\mathcal{H}(X)$ lying between $\mathrm{CL}(X)$ and $\mathcal{F}(X)$. For infinite metrizable spaces, we characterize several cases in which $\mathsf{SC}(X)$ and $\mathsf{SC}(\mathrm{CL}(X))$ differ and establish an independence result over \textsf{ZFC}; see Theorem~5.12.

math.GN↗

arXiv2026

Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces

For infinite cardinals $λ\leq κ$, we calculate the calibers of topological spaces $X$ of cardinality $κ$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < λ$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.

math.GN↗

arXiv2024

Remarks on SHD spaces and more divergence properties

The class of SHD spaces was recently introduced in [12]. The first part of this paper focuses on answering most of the questions presented in that article. For instance, we exhibit an example of a non-SHD Tychonoff space $X$ such that $\mathscr{F}[X]$, the Pixley-Roy hyperspace of $X$, $βX$, the Stone-Čech compactification of $X$, and $C_p(X)$, the ring of continuous functions over $X$ equipped with the topology of pointwise convergence, are SHD. In the second part of this work we will present some variations of the SHD notion, namely, the WSHD property and the OHD property. Furthermore, we will pay special attention to the relationships between $X$ and $\mathscr{F}[X]$ regarding these new concepts.

math.GN↗

arXiv2024

On calibers for $C_p(X)$

We present new results regarding calibers in the function spaces $C_p(X)$. Our main theorem is that $C_p(X)$ is strongly Šanin whenever $X$ is a submetrizable space; this improves an earlier result due to Tkachuk: $C_p(X)$ is Šanin whenever $X$ is a submetrizable space. Moreover, we give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{ξ< λ}X_ξ$ is a product of non-trivial Tychonoff spaces with $i$-weight $\leq λ$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $λ$-Lindelöf extension of a discrete space of cardinality $\geq λ$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)=κ^{+}$ and $C_p(Z^κ)$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_α: α<cf(κ)\}$ such that $κ$ is a caliber for $C_p(Z_α)$ whenever $α<cf(κ)$ but it is not a caliber for $C_p(\bigoplus_{α<cf(κ)} Z_α)$.

math.GN↗

arXiv2023

On selectively highly divergent spaces

We say that a topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open sets $\{U_n\mid n\inω\}$ of $X$, we can find $x_n\in U_n$ such that the sequence $(x_n)$ has no convergent subsequences. We investigate the basic topological properties of SHD spaces and we will exhibit that this class of spaces is full of variety. We present an example of a SHD space wich has a non trivial convergent sequence and with a dense set with no convergent sequences. Also, we prove that if $X$ is a regular space such that for all $x\in X$ holds $ψ(x,X)>ω$, then $X_δ$ (the $G_δ$ modification of $X$) is a SHD space and, moreover, if $X$ homogeneous, then $X_δ$ is also homogeneous. Finally, given $X$ a Hausdorff space without isolated points, we construct a new space denoted by $sX$ such that $sX$ is extremally disconnected, zero-dimensional Hausdorff space, SHD with $|X|=|sX|$, $πw(X)=πw(sX)$ and $c(X)=c(sX)$ where $πw$ and $c$ are the cardinal functions $π$-weight and celullarity respectively.

math.GN↗

arXiv2023

Some notes on topological calibers

We show that the definition of caliber given by Engelking in R. Engelking, "General topology", Sigma series in pure mathematics, Heldermann, vol. 6, 1989, which we will call caliber*, differs from the traditional notion of this concept in some cases and agrees in others. For instance, we show that if $κ$ is an infinite cardinal with $2^κ<\aleph_κ$ and $cf(κ)>ω$, then there exists a compact Hausdorff space $X$ such that $o(X)=2^{\aleph_κ}=|X|$, $\aleph_κ$ is a caliber* for $X$ and $\aleph_κ$ is not a caliber for $X$. On the other hand, we obtain that if $λ$ is an infinite cardinal number, $X$ is a Hausdorff space with $|X|>1$, $ϕ\in \{w ,nw\}$, $o(X) = 2^{ϕ(X)}$ and $μ:= o\left(X^λ\right)$, then the calibers of $X^λ$ and the true calibers* (that is, those which are less than or equal to $μ$) coincide, and are precisely those that have uncountable cofinality.

math.GN↗

arXiv2023

Weakly separated spaces and Pixley-Roy hyperspaces

In this paper we obtain new results regarding the chain conditions in the Pixley-Roy hyperspaces $\mathscr{F}[X]$. For example, if $c(X)$ and $R(X)$ denote the cellularity and weak separation number of $X$ (see Section~[4]) and we define the cardinals $$c^* (X) := \sup \{c(X^{n}) : n\in \mathbb{N}\} \quad \text{and} \quad R^{*}(X) := \sup \{R(X^{n}) : n\in \mathbb{N}\},$$ then we show that $R^{*}(X) = c^ {*}\left(\mathscr{F}[X]\right)$. On the other hand, in "M. Sakai, Cardinal functions of Pixley-Roy hyperspaces, Topol. Appl., 159 (2012), 3080--3088." Sakai asked whether the fact that $\mathscr{F}[X]$ is weakly Lindelöf implies that $X$ is hereditarily separable and proved that if $X$ is countably tight then the previous question has an affirmative answer. We shall expand Sakai's result by proving that if $\mathscr{F}[X]$ is weakly Lindelöf and $X$ is a Hausdorff $k$-space; or $X$ is a countably tight $T_1$ space; or $X$ is weakly separated, then $X$ is hereditarily separable.

math.GN↗

arXiv2023

New results regarding the lattice of uniform topologies on $C(X)$

For a Tychonoff space $X$, the lattice $\mathscr U_X$ was introduced in L.A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, "The lattice of uniform topologies on $C(X)$", Questions and Answers in General Topology, 39 (2021), 65-71. In the present paper we continue the study of $\mathscr U_X$. To be specific, the present paper deals, in its first half, with structural and categorical properties of $\mathscr U_X$, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space $X$.

math.GN↗

arXiv2022

First-countable Lindelöf scattered spaces

We study the class of first-countable Lindelöf scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a scattered subspace of $\mathbb Q$, the class of $T_2$ FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to $Q$-sets, Lusin sets, and their relatives, and to the cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Many natural questions about FLS spaces turn out to be independent of $\mathsf{ZFC}$. We prove that there exist uncountable FLS spaces with scattered height $ω$. On the other hand, an uncountable FLS space with finite scattered height exists if and only if $\mathfrak{b} = \aleph_1$. We prove some independence results concerning the possible cardinalities of FLS spaces, and concerning what ordinals can be the scattered height of an FLS space. Several open problems are included.

math.GN↗

arXiv2022

Singular precalibers for topological products

In this paper we prove that if $κ$ is a singular cardinal with uncountable cofinality, then every power of a given topological space with precaliber $κ$ has precaliber $κ$ as well. Furthermore, if $\{X_α: α<λ\}$ is a family of topological spaces with precaliber $κ$ and $λ<cf(κ)$, then $κ$ is a precaliber for the topological product $\prod\{X_α: α<λ\}$.

math.GN↗