arXiv subjects

Davide Giacopello

Publications and source records attributed to Davide Giacopello.

8 recordsLinked to original sources

arXiv2026

A topological characterization of end space of infinite graphs via games, subspaces and products

In 1992, Diestel asked which topological spaces could be represented as the end space of some graph. In 2023, Pitz provided a solution to this question by giving a topological characterization of end spaces using a hereditarily complete special subbase. In this paper, we present an alternative topological characterization of end spaces, in which we employ a special subbase and a topological game. Furthermore, we provide several applications of this characterization: we show that every end space is hereditarily Baire, that $G_δ$ subspaces of end spaces are also end spaces, and that the product of end spaces is not always an end space.

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arXiv2025

Totally paracompact spaces and the Menger covering property

A topological space is totally paracompact if any base of this space contains a locally finite subcover. We focus on a problem of Curtis whether in the class of regular Lindelöf spaces total paracompactness is equivalent to the Menger covering property. To this end we consider topological spaces with certain dense subsets. It follows from our results that the above equivalence holds in the class of Lindelöf GO-spaces defined on subsets of reals. We also provide a game-theoretical proof that any regular Menger space is totally paracompact and show that in the class of first-countable spaces the Menger game and a partial open neighborhood assignment game of Aurichi are equivalent. We also show that if $\mathfrak{b}=ω_1$, then there is an uncountable subspace of the Sorgenfrey line whose all finite powers are Lindelöf, which is a strengthening of a famous result due to Michael.

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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

On spaces with a $π$-base whose elements have an H-closed closure

We deal with the class of Hausdorff spaces having a $π$-base whose elements have an H-closed closure. Carlson proved that $|X|\leq 2^{wL(X)ψ_c(X)t(X)}$ for every quasiregular space $X$ with a $π$-base whose elements have an H-closed closure. We provide an example of a space $X$ having a $π$-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that $|X|> 2^{wL(X)χ(X)}$ (then $|X|> 2^{wL(X)ψ_c(X)t(X)}$). Still in the class of spaces with a $π$-base whose elements have an H-closed closure, we establish the bound $|X|\leq2^{wL(X)k(X)}$ for Urysohn spaces and we give an example of an Urysohn space $Z$ such that $k(Z)<χ(Z)$. Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a $π$-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a $π$-base whose elements have an H-closed closure then such space is Baire.

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arXiv2023

On some topological games involving networks

In these notes we introduce and investigate two new games called R-nw-selective game and the M-nw-selective game. These games naturally arise from the corresponding selection principles involving networks introduced in \cite{BG}.

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arXiv2023

New bounds on the cardinality of n-Hausdorff and n-Urysohn spaces

Two new cardinal functions defined in the class of $n$-Hausdorff and $n$-Urysohn spaces that extend pseudocharacter and closed pseudocharacter respectively are introduced. Through these new functions bounds on the cardinality of $n$-Urysohn spaces that represent variations of known results are given. Also properties of $n$-Urysohn $n$-H-closed spaces are proved.

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arXiv2023

On n-Hausdorff homogeneous and n-Urysohn homogeneous spaces

In this paper we study $n$-Hausdorff homogeneous and $n$-Urysohn homogeneous spaces. We give some upper bounds for the cardinality of these kind of spaces and give examples. Additionally we show that for every $n>2$, there is no $n$-Hausdorff 2-homogeneous space. Finally, for any $n$-Hausdorff space we construct an $n$-Hausdorff homogeneous extension which is the union of countably many $n$-H-closed spaces.

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arXiv2022

Some properties defined by relative versions of star-covering properties II

In this paper we consider some recent relative versions of Menger property called set strongly star Menger and set star Menger properties and the corresponding Hurewicz-type properties. In particular, using \cite {BMae}, we "easily" prove that the set strong star Menger and set strong star Hurewicz properties are between countable compactness and the property of having countable extent. Also we show that the extent of a regular set star Menger or a set star Hurewicz space cannot exceed $\frak c$. Moreover, we construct (1) a consistent example of a set star Menger (set star Hurewicz) space which is not set strongly star Menger (set strongly star Hurewicz) and show that (2) the product of a set star Menger (set star Hurewicz) space with a compact space need not be set star Menger (set star Hurewicz). In particular, (1) and (2) answer to some questions posed by Kočinac, Konca and Singh in \cite{KKS-MS} and \cite{S-AM}.

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