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Rodrigo Hernández-Gutiérrez

Publications and source records attributed to Rodrigo Hernández-Gutiérrez.

At least 19 recordsLinked to original sources
arXiv2026

Functional countability and exponential separability of product spaces and subspaces

We investigate the behavior of functional countability and exponential separability in products and subspaces of topological spaces. We solve a problem of Tkachuk by showing that the product of functionally countable pseudocompact spaces is itself functionally countable. Solving another problem of Tkachuk, we show that it is independent of ZFC whether regular spaces which have all their subspaces functionally countable are hereditarily Lindelöf. Finally, we prove that the $σ$-product of non-zero ordinals is exponentially separable, thereby extending a result of Kemoto and Szeptycki.

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arXiv2024

Comparing functional countability and exponential separability

A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponentially separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable and the influence of the $G_δ$-topology on exponential separability. Finally, we give some examples of functionally countable spaces that are separable and uncountable.

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arXiv2024

Consistency and independence phenomena involving cellular-Lindelof spaces

The cellular-Lindelöf property is a common generalization of the Lindelöf property and the countable chain condition that was introduced by Bella and Spadaro in 2018. We solve two questions of Alas, Gutierrez-Dominguez and Wilson by constructing consistent examples of a normal almost cellular-Lindelöf space which is neither cellular-Lindelöf nor weakly Lindelöf and a Tychonoff cellular-Lindelöf space of Lindelöf degree $ω_1$ and uncountable weak Lindelöf degree for closed sets. We also construct a ZFC example of a space for which both the cellular-Lindelöf property and normality are undetermined in ZFC.

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arXiv2024

When is the complement of the diagonal of a LOTS functionally countable?

In a 2021 paper, Vladimir Tkachuk asked whether there is a non-separable LOTS $X$ such that $X^2\setminus\{\langle x,x\rangle\colon x\in X\}$ is functionally countable. In this paper we prove that such a space, if it exists, must be an Aronszajn line and admits a $\leq 2$-to-$1$ retraction to a subspace that is a Suslin line. After this, assuming the existence of a Suslin line, we prove that there is Suslin line that is functionally countable. Finally, we present an example of a functionally countable Suslin line $L$ such that $L^2\setminus\{\langle x,x\rangle\colon x\in L\}$ is not functionally countable.

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arXiv2023

Smooth fans that are endpoint rigid

Let $X$ be a smooth fan and denote its set of endpoints by $E(X)$. Let $E$ be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan $X$ such that $E(X)$ is homeomorphic to $E$ and for every homeomorphism $h \colon X \to X$, the restriction of $h$ to $E(X)$ is the identity. On the other hand, we also prove that if $X$ is any smooth fan such that $E(X)$ is homeomorphic to complete Erdős space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik.

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arXiv2023

The hyperspace of non-cut subcontinua of graphs and dendrites

Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)=\{ A \in C(X):X\setminus A\text{ is connected}\}$ when $X$ is a finite graph or a dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact, connected, locally connected or totally disconnected. Also, we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.

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arXiv2022

A characterization of the product of the rational numbers and complete Erdős space

Erdős space $\mathfrak{E}$ and complete Erdős space $\mathfrak{E}_c$ have been previously shown to have topological characterizations. In this paper, we provide a topological characterization of the topological space $\mathbb{Q}\times\mathfrak{E}_c$, where $\mathbb{Q}$ is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets $\mathcal{F}(\mathfrak{E}_c)$ is homeomorphic to $\mathbb{Q}\times\mathfrak{E}_c$. We also characterize the factors of $\mathbb{Q}\times\mathfrak{E}_c$. An interesting open question that is left open is whether $σ{\mathfrak{E}_c}^ω$, the $σ$-product of countably many copies of $\mathfrak{E}_c$, is homeomorphic to $\mathbb{Q}\times\mathfrak{E}_c$.

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arXiv2020

Countable dense homogeneity of function spaces

In this paper we consider the question of when the space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology is countable dense homogeneous. In particular, we focus on the case when $X$ is countable with a unique non-isolated point $\infty$. In this case, $C_p(X)$ is countable dense homogeneous if and only if the filter of open neighborhoods of $\infty$ is a non-meager $P$-filter.

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arXiv2019

Pseudoradial spaces and copies of $ω_1+1$

In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that $\mathrm{MA}+\mathfrak{c}=ω_2$ implies that there is a compact pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space $X$ and a continuous function $f:X\toω_1+1$ in such a way that there is no copy of $ω_1+1$ in $X$ that maps cofinally under $f$. We also give some conditions that imply the existence of copies of $ω_1$ in spaces. In particular, $\mathrm{PFA}$ implies that compact almost radial spaces of radial character $ω_1$ contain many copies of $ω_1$.

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arXiv2018

Far points and discretely generated spaces

We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the cardinal number given by the smallest possible character of remote and far sets of separable metrizable spaces. Finally, we prove that in some cases a countable space has far points.

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arXiv2018

Disconnectedness properties of Hyperspaces

Let $X$ be a Hausdorff space and let $\mathcal{H}$ be one of the hyperspaces $CL(X)$, $\mathcal{K}(X)$, $\mathcal{F}(X)$ or $\mathcal{F}_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathcal{H}$: extremal disconnectedness, being a $F^\prime$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that $(a)$ both $F$ and $X-F$ are totally disconnected, $(b)$ the quotient $X/F$ is hereditarily disconnected, then $\mathcal{K}(X)$ is hereditarily disconnected. We also show an example proving that this result cannot be reversed.

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arXiv2018

Spaces of Remote Points

Given a Tychonoff space $X$, let $\varrho(X)$ be the set of remote points of $X$. We view $\varrho(X)$ as a topological space. In this paper we assume that $X$ is metrizable and ask for conditions on $Y$ so that $\varrho(X)$ is homeomorphic to $\varrho(Y)$. This question has been studied before by R. G. Woods and C. Gates. We give some results of the following type: if $X$ has topological property $\mathbf{P}$ and $\varrho(X)$ is homeomorphic to $\varrho(Y)$, then $Y$ also has $\mathbf{P}$. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that $\varrho(X)$ and $\varrho(Y)$ have open dense homeomorphic subspaces if $X$ and $Y$ are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.

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arXiv2018

Countable Dense Homogeneity and the Double Arrow Space

Let $\mathbb{A}$ denote the Alexandroff-Urysohn double arrow space. We prove the following results: (a) $\mathbb{A}\times{}^ω{2}$ is not countable dense homogeneous; (b) ${}^ω{\mathbb{A}}$ is not countable dense homogeneous; (c) $\mathbb{A}$ has exactly $\mathfrak{c}$ types of countable dense subsets. These results answer questions by Arhangel'ski\uı, Hrušák and van Mill.

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arXiv2018

Countable dense homogeneity and $λ$-sets

We show that all sufficiently nice $λ$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $κ\le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $κ$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $κ$ is equivalent to the existence of a $λ$-set of size $κ$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $ω_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable $\mathsf{CDH}$ space in ZFC.

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arXiv2018

Dendrites and symmetric products

For a given continuum $X$ and a natural number $n,$ we consider the hyperspace $F_n(X)$ of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. In this paper we show that if $X$ is a dendrite whose set of end points is closed, $n \in \mathbb{N}$ and $Y$ is a continuum such that the hyperspaces $F_n(X)$ and $F_n(Y)$ are homeomorphic, then $Y$ is a dendrite whose set of end points is closed.

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arXiv2018

A non-discrete space $X$ with $C_p(X)$ Menger at infinity

In a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space $X$ such that the remainder of $C_p(X)$ in some compactification is Menger but not $σ$-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.

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arXiv2017

Countable dense homogeneity and the Cantor set

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $π$-weight, if ${}^ω{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.

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