Related papers: On the cardinality of Hausdorff spaces
We present a bound for the weak Lindel\"of number of the $G_\delta$-modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in…
We prove that, under CH, any space with a regular $G_\delta$-diagonal and caliber $\omega_1$ is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space $X$, we establish the inequality $|X|\le…
Much recent work in cardinal characteristics has focused on generalizing results about $\omega$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$…
Combining ideas from two of our previous papers, we refine Arhangel'skii Theorem by proving a cardinal inequality of which this is a special case: any increasing union of strongly discretely Lindelof spaces with countable free sequences and…
Given an open cover $\mathcal{U}$ of a topological space $X$, we introduce the notion of a star network for $\mathcal{U}$. The associated cardinal function $sn(X)$, where $e(X)\leq sn(X)\leq L(X)$, is used to establish new cardinal…
Arhangel'skii proved that if a first countable Hausdorff space is Lindel\"of, then its cardinality is at most $2^{\aleph_0}$. Such a clean upper bound for Lindel\"of spaces in the larger class of spaces whose points are ${\sf G}_{\delta}$…
We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq…
The inequality $|X| \leq 2^{\chi(X)}$ has been proved to be true for Lindel\"of spaces (Arhangel'ski\u\i, 1969), $H$-closed spaces (Dow-Porter, 1982) and ccc spaces (Hajnal-Ju\'asz 1967), by quite different arguments. We present a common…
We affirmatively solve the main problems posed by Laczkovich and Paulin in \emph{Stability constants in linear spaces}, Constructive Approximation 34 (2011) 89--106 (do there exist cases in which the second Whitney constant is finite while…
We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular…
A generalization of the H\"older inequality is considered. Its relations with a previously obtained improvement of the Cauchy--Schwarz inequality are discussed.
In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $\kappa$-Borel spaces for $\kappa$ an uncountable (regular) cardinal satisfying $\kappa^{<\kappa}…
Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable…
We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric…
In this article, using ideas of Liero, Mielke and Savar\'{e} in [21], we establish a Kantorovich duality for generalized Wasserstein distances $W_1^{a,b}$ on a generalized Polish metric space, introduced by Picolli and Rossi. As a…
In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky's conjecture [16] and…
The notion of Hausdorff number of a topological space is first introduced in \cite{bonan}, with the main objective of using this notion to obtain generalizations of some known bounds for cardinality of topological spaces. Here we consider…
We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact…
We show, in a certain specific sense, that both the density and the cardinality of a Hausdorff space are related to the "degree" to which the space is nonregular. It was shown by Sapirovskii that $d(X)\leq\pi\chi(X)^{c(X)}$ for a regular…
The notion of a $\Gamma $-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group $\Gamma $ replaces the group $Z_2$. The case $\Gamma =\Z_k$ has also been studied, from the…