arXiv subjects

Santi Spadaro

Publications and source records attributed to Santi Spadaro.

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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arXiv2025

On (non-Menger) spaces whose closed nowhere dense subsets are Menger

A space $X$ is od-Menger if it satisfies $\mathsf{U_{fin}}(Δ_X, \mathcal{O}_X)$, where $\mathcal{O}_X,Δ_X$ are the collection of covers of $X$ by respectively open subsets and open dense subsets. We show that under CH, there is a refinement of the usual topology on a subset of the reals which yields a hereditarily Lindelöf, od-Menger, non-Menger, $0$-dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.

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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

On some recent selective properties involving networks

In this paper we investigate R-,H-, and M-{\it nw}-selective properties introduced in \cite{BG}. In particular, we provide consistent uncountable examples of such spaces and we define \textit{trivial} R-,H-, and M-{\it nw}-selective spaces the ones with countable net weight having, additionally, the cardinality and the weight strictly less then $cov({\cal M})$, $\frak b$, and $\frak d$, respectively. Since we establish that spaces having cardinalities more than $cov({\cal M})$, $\frak b$, and $\frak d$, fail to have the R-,H-, and M-{\it nw}-selective properties, respectively, non-trivial examples should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-{\it nw}-selective and H-{\it nw}-selective spaces and we establish some limitations to constructions of non-trivial examples. Moreover, we consistently prove the existence of two H-{\it nw}-selective spaces whose product fails to be M-{\it nw}-selective. Finally, we study some relations between {\it nw}-selective properties and a strong version of the HFD property.

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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

Strongly discrete subsets with Lindelöf closures

We define a topological space to be an "SDL space" if the closure of each one of its strongly discrete subsets is Lindelöf. After distinguishing this property from the Lindelöf property we make various remarks about cardinal invariants of SDL spaces. For example we prove that $|X| \leq 2^{χ(X)}$ for every SDL Urysohn space and that every SDL $P$-space of character $\leq ω_1$ is regular and has cardinality $\leq 2^{ω_1}$. Finally, we exploit our results to obtain some partial answers to questions about the cardinality of cellular-Lindelöf spaces.

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arXiv2024

Dense metrizable subspaces in powers of Corson compacta

We characterize when the countable power of a Corson compactum has a dense metrizable subspace and construct consistent examples of Corson compacta whose countable power does not have a dense metrizable subspace. We also give several remarks about ccc Corson compacta and, as a byproduct, we obtain a new proof of Kunen and van Mill's characterization of when a Corson compactum supporting a strictly positive measure is metrizable.

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arXiv2024

A regular non-weakly discretely generated P-space

We construct a consistent example of a topological space $Y=X \cup \{\infty\}$ such that: 1) $Y$ is regular. 2) Every $G_δ$ subset of $Y$ is open. 3) The point $\infty$ is not isolated, but it is not in the closure of any discrete subset of $X$.

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arXiv2023

On two questions on selectively highly divergent spaces

A topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open subsets $\{U_n: n\in ω\}$ of $X$, we can pick a point $x_n\in U_n$, for every $n<ω$, such that the sequence $\{x_n: n\inω\} $ has no convergent subsequences. In this note we answer four questions related to this notion asked in ArXiv:2307.11992.

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arXiv2020

Free sequences and the tightness of pseudoradial spaces

Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial character of every Lindelöf Hausdorff almost radial space $X$ and the set-tightness of every Lindelöf Hausdorff space are always bounded above by $F(X)$. Solving a question of Bella, we exhibit a Hausdorff radial space $X$ whose radial character is strictly larger than $F(X)$. We then improve a result of Dow, Juhász, Soukup, Szentmiklóssy and Weiss by proving that if $X$ is a Lindelöf Hausdorff space, and $X_δ$ denotes the $G_δ$ topology on $X$ then $t(X_δ) \leq 2^{t(X)}$. Finally, we exploit this to prove that if $X$ is a Lindelöf Hausdorff pseudoradial space then $F(X_δ) \leq 2^{F(X)}$, which partially answer a question of Bella and ourselves.

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arXiv2020

Upper bounds for the tightness of the $G_δ$-topology

We prove that if $X$ is a regular space with no uncountable free sequences, then the tightness of its $G_δ$ topology is at most continuum and if $X$ is in addition Lindelöf then its $G_δ$ topology contains no free sequences of length larger then the continuum. We also show that the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than $ω_1$, but whose $G_δ$ topology can have arbitrarily large tightness.

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arXiv2019

A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality

We present a bound for the weak Lindelöf number of the $G_δ$-modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\le 2^{L(X)χ(X)}$ (Arhangel'skii) and $|X|\le 2^{c(X)χ(X)}$ (Hajnal-Juhasz). This solves a question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not need any separation axiom beyond $T_2$.

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arXiv2019

Cardinal invariants of cellular-Lindelof spaces

A space $X$ is said to be "cellular-Lindelöf" if for every cellular family $\mathcal{U}$ there is a Lindelöf subspace $L$ of $X$ which meets every element of $\mathcal{U}$. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelöf monotonically normal space is Lindelöf and that every cellular-Lindelöf space with a regular $G_δ$-diagonal has cardinality at most $2^\mathfrak{c}$. We also prove that every normal cellular-Lindelöf first-countable space has cardinality at most continuum under $2^{<\mathfrak{c}}=\mathfrak{c}$ and that every normal cellular Lindelöf space with a $G_δ$-diagonal of rank $2$ has cardinality at most continuum.

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arXiv2018

On closures of discrete sets

The depth of a topological space $X$ ($g(X)$) is defined as the supremum of the cardinalities of closures of discrete subsets of $X$. Solving a problem of Martínez-Ruiz, Ramírez-Páramo and Romero-Morales, we prove that the cardinal inequality $|X| \leq g(X)^{L(X) \cdot F(X)}$ holds for every Hausdorff space $X$, where $L(X)$ is the Lindelöf number of $X$ and $F(X)$ is the supremum of the cardinalities of the free sequences in $X$.

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arXiv2018

A note on rank 2 diagonals

We solve two questions regarding spaces with a ($G_δ$)-diagonal of rank 2. One is a question of Basile, Bella and Ridderbos regarding weakly Lindelöf spaces with a $G_δ$-diagonal of rank 2 and the other is a question of Arhangel'skii and Bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum.

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arXiv2017

On the cardinality of almost discretely Lindelof spaces

A space is said to be "almost discretely Lindelöf" if every discrete subset can be covered by a Lindelöf subspace. Juhász, Tkachuk and Wilson asked whether every almost discretely Lindelöf first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $2^{<\mathfrak{c}}=\mathfrak{c}$ (which is a consequence of Martin's Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhász, Soukup and Szentmiklóssy. We conclude with a few related results and questions.

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arXiv2017

$G_δ$ covers of compact spaces

We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_δ$ cover with no continuum-sized ($G_δ$)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every $G_δ$ cover has a $\mathfrak{c}$-sized subcollection with a $G_δ$-dense union and that in a Lindelöf space with a base of multiplicity continuum, every $G_δ$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.

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arXiv2017

Cardinal Invariants for the $G_δ$ topology

We prove upper bounds for the spread, the Lindelöf number and the weak Lindelöf number of the $G_δ$-topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juhász and van Mill regarding the cardinality of a $σ$-countably tight homogeneous compactum.

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