Some results on cardinality bounds and covering-type properties of a topological space.
type: CreativeWork
description: In this study we consider cardinal functions in topology with a primary focus on cardinality bounds, and covering properties. In the first chapter of this thesis we present results on bounds for the cardinality of a topological space using cardinal functions. In particular, we give variations and improvements of the most known cardinality bounds for topological spaces. -In Section 1.1 we pose our attention on cardinality bounds for Urysohn spaces. In our work we focused on both Schroder and Sapirovskii's inequalities. Schroder proved that if $X$ is a Urysohn space, then $|X|\leq 2^{Uc(X)\chi(X)}$, and Sapirovskii proved that if $X$ is a regular space, then $|X|\leq\pi_{\chi}(X)^{c(X)\psi(X)}$. We define, in the class of Urysohn spaces, a new cardinal function called $\theta$-pseudocharacter of a space to prove the following: If $X$ is a Urysohn space, then $|X|\leq \pi\chi(X)^{Uc(X)\psi_{\theta}(X)}$. This is a common generalization of both the Schroder's inequality and Sapirovskii's inequality mentioned above. Other cardinality bounds for Urysohn spaces are proved, including an improvement of the Bella-Cammaroto inequality. -In Section 1.2 we introduce a new cardinal invariant called quasicellularity of a space in order to prove cardinality restrictions for Hausdorff spaces. The second chapter of this thesis is dedicated to results obtained in the field of covering properties for topological spaces. Lots of important theorems in general topology make use of covering properties. In particular we consider covering properties defined by stars, neighborhood assignments or as monotone versions of selection principles. Star covering properties have been widely studied in literature and the use of stars is very important in general topology. In fact, some topological and covering properties are characterized using stars. Recall that if $A$ is a subset of a space $X$ and ${\mathcal B}$ is a family of subsets of $X$, the star of $A$ with respect to ${\mathcal B}$, denoted b…
abstract: In this study we consider cardinal functions in topology with a primary focus on cardinality bounds, and covering properties. In the first chapter of this thesis we present results on bounds for the cardinality of a topological space using cardinal functions. In particular, we give variations and improvements of the most known cardinality bounds for topological spaces. -In Section 1.1 we pose our attention on cardinality bounds for Urysohn spaces. In our work we focused on both Schroder and Sapirovskii's inequalities. Schroder proved that if $X$ is a Urysohn space, then $|X|\leq 2^{Uc(X)\chi(X)}$, and Sapirovskii proved that if $X$ is a regular space, then $|X|\leq\pi_{\chi}(X)^{c(X)\psi(X)}$. We define, in the class of Urysohn spaces, a new cardinal function called $\theta$-pseudocharacter of a space to prove the following: If $X$ is a Urysohn space, then $|X|\leq \pi\chi(X)^{Uc(X)\psi_{\theta}(X)}$. This is a common generalization of both the Schroder's inequality and Sapirovskii's inequality mentioned above. Other cardinality bounds for Urysohn spaces are proved, including an improvement of the Bella-Cammaroto inequality. -In Section 1.2 we introduce a new cardinal invariant called quasicellularity of a space in order to prove cardinality restrictions for Hausdorff spaces. The second chapter of this thesis is dedicated to results obtained in the field of covering properties for topological spaces. Lots of important theorems in general topology make use of covering properties. In particular we consider covering properties defined by stars, neighborhood assignments or as monotone versions of selection principles. Star covering properties have been widely studied in literature and the use of stars is very important in general topology. In fact, some topological and covering properties are characterized using stars. Recall that if $A$ is a subset of a space $X$ and ${\mathcal B}$ is a family of subsets of $X$, the star of $A$ with respect to ${\mathcal B}$, denoted b…
author: BASILE, FORTUNATA AURORA
language: ita
keywords: general topology, cardinality restrictions, cardinal functions, covering properties
date published: 2020-03-27
date modified: 2026-07-24T01:35:01Z
provider: Università degli Studi di Catania
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Università degli studi di Catania

PropertyValue
property ID: Handle
value: https://hdl.handle.net/20.500.11769/581795