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.
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.
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.
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}.
On a Weaker Form of Countable Compactness Bonanzinga, Maddalena; Cammaroto, Filippo; Matveev, Mikhail V.
Quaestiones mathematicae,
12/2007, Letnik:
30, Številka:
4
Journal Article
Recenzirano
A star covering property which is equivalent to countable compactness for regular spaces and weaker than countable compactness for Hausdorff spaces is introduced and considered. Various kinds of ...irregularity of topological spaces are discussed.
On Weaker Forms of Separability Bonanzinga, Maddalena; Cammaroto, Filippo; Matveev, Mikhail ...
Quaestiones mathematicae,
12/1/2008, 2008-12-00, Letnik:
31, Številka:
4
Journal Article
Recenzirano
Two properties each of which is equivalent to separability for regular spaces and is weaker than separability in general are considered as well as the corresponding cardinal functions.
Some variations of the Banach-Mazur game Leandro Fiorini Aurichi; Bonanzinga, Maddalena; Gabriel Andre Asmat Medina
arXiv (Cornell University),
04/2021
Paper, Journal Article
Odprti dostop
The classical Banach-Mazur game is directly related to the Baire property and the property of being a productively Baire space. In this paper, we discuss two variations of this classic game that are ...even more related to these properties.
In this paper we extend the theory of H-closed extensions of Hausdorff spaces
to a class of non-Hausdorff spaces, defined in \cite{B}, called $n$-Hausdorff
spaces. The notion of H-closed is ...generalized to an $n$-H-closed space. Known
construction for Hausdorff spaces $X$, such as the Kat\v{e}tov H-closed
extension $\kappa X$, are generalized to a maximal $n$-H-closed extension
denoted by $n$-$\kappa X$.
Sapirovskii 18 proved that \(|X|\leq\pi\chi(X)^{c(X)\psi(X)}\), for a regular space \(X\). We introduce the \(\theta\)-pseudocharacter of a Urysohn space \(X\), denoted by \(\psi_\theta (X)\), and ...prove that the previous inequality holds for Urysohn spaces replacing the bounds on celluarity \(c(X)\leq\kappa\) and on pseudocharacter \(\psi(X)\leq\kappa\) with a bound on Urysohn cellularity \(Uc(X)\leq\kappa\) (which is a weaker conditon because \(Uc(X)\leq c(X)\)) and on \(\theta\)-pseudocharacter \(\psi_\theta (X)\leq\kappa\) respectivly (note that in general \(\psi(\cdot)\leq\psi_\theta (\cdot)\) and in the class of regular spaces \(\psi(\cdot)=\psi_\theta(\cdot)\)). Further, in 6 the authors generalized the Dissanayake and Willard's inequality: \(|X|\leq 2^{aL_{c}(X)\chi(X)}\), for Hausdorff spaces \(X\) 25, in the class of \(n\)-Hausdorff spaces and de Groot's result: \(|X|\leq 2^{hL(X)}\), for Hausdorff spaces 11, in the class of \(T_1\) spaces (see Theorems 2.22 and 2.23 in 6). In this paper we restate Theorem 2.22 in 6 in the class of \(n\)-Urysohn spaces and give a variation of Theorem 2.23 in 6 using new cardinal functions, denoted by \(UW(X)\), \(\psi w_\theta(X)\), \(\theta\hbox{-}aL(X)\), \(h\theta\hbox{-}aL(X)\), \(\theta\hbox{-}aL_c(X)\) and \(\theta\hbox{-}aL_{\theta}(X)\). In 5 the authors introduced the Hausdorff point separating weight of a space \(X\) denoted by \(Hpsw(X)\) and proved a Hausdorff version of Charlesworth's inequality \(|X|\leq psw(X)^{L(X)\psi(X)}\) 7. In this paper, we introduce the Urysohn point separating weight of a space \(X\), denoted by \(Upsw(X)\), and prove that \(|X|\leq Upsw(X)^{\theta\hbox{-}aL_{c}(X)\psi(X)}\), for a Urysohn space \(X\).
