Some results on cardinality bounds and covering-type properties of a topological space.

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 by $St(A,{\mathcal B})$, is the set $\bigcup\{B\in{\mathcal B}:B\cap A\neq\emptyset\}$. It is natural to consider stars of subspaces having particular properties. In this case we will say that a space $X$ has the star-$\mathcal{P}$ property if for every open cover $\mathcal{U}$ of the space $X$, there exists a subset $Y$ of $X$ having the property $\mathcal{P}$ such that $St(Y,{\mathcal U})=X$. We can also consider covering properties not only in terms of stars but also in terms of neighborhood assignments. Recall that a neighborhood assignment in a space $X$ is a family $\{O_x:x\in X\}$ of open subsets of $X$ such that $x\in O_x$ for every $x\in X$. For example the Lindelof property can be characterized using neighborhood assignments in the following way: a space $X$ is Lindelof if and only if for every neighborhood assignment $\{O_x:x\in X\}$ there is a countable subset $Y$ of $X$ such that $\{O_x:x\in Y\}$ is a cover of $X$. If $Y$ is closed and discrete instead of countable we obtain the notion of D-spaces defined by van Douwen. The idea of van Douwen have been generalized by van Mill, Tkachuk, and Wilson. They defined a space to be neighborhood assignment $\mathcal P$ if for any neighborhood assignment $\{O_x:x\in X\}$ there exists a subspace $Y$ of $X$ having the property $\mathcal P$ such that $\{O_x:x\in Y\}$ is a cover of $X$. -In Section 2.1 we study and compare "star" and "neighborhood assignment" versions of compactness, countable compactness, Lindelofness, and of the Menger property. -In Section 2.2 we consider monotone versions of some selection principles. When we add monotonicity to a covering property, we obtain a stronger property. The idea of a covering property being monotonic has its roots in the definition of "monotone normality" that has nothing to do with open covers. A space $X$ is called monotonically normal if for each pair $(H,K)$ of disjoint closed subsets of $X$, one can assign an open set $r(H,K)$ such that $H\subset r(H, K)\subset\overline{r(H, K)}\subset X\setminus K$, and if $H_1\subset H_2$ and $K_1\supset K_2$ then $r(H_1, K_1)\subset r(H_2, K_2)$. Shortly after, the style of this definition was adapted and applied to other kinds of properties, including covering properties. Gartside and Moody described a process for obtaining a monotone version of any well-known covering property: “by requiring that there is an operator, $r$, assigning to every open cover a refinement in such a way that $r(\mathcal V )$ refines $r(\mathcal U )$ whenever $\mathcal V$ refines $\mathcal U$ ”. Using this process, any covering property can be "upgrated" into a monotonic property. We study and compare the four different ways of defining monotone versions of Menger, Rothberger and Hurewicz properties, and we show that one of this monotone versions introduced is absurd.