Insights into Topological Spaces: Bounds on the Cardinality of Spaces, Selection Principles involving Networks, and related Games

In Chapter \ref{Chapter1} we present the theory of cardinal invariants and the research in cardinal upper bounds of topological spaces. Then, we deal with the class of Hausdorff spaces having a $\pi$-base whose elements have an H-closed closure. In 2023, Nathan Carlson proved that $|X|\leq 2^{wL(X)\psi_c(X)t(X)}$ for every quasiregular space $X$ with a $\pi$-base whose elements have an H-closed closure. We provide an example of a space $X$ having a $\pi$-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that $|X|> 2^{wL(X)\chi(X)}$ (then $|X|> 2^{wL(X)\psi_c(X)t(X)}$). In the class of spaces with a $\pi$-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 $k(Z)<\chi(Z)$. Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a $\pi$-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a $\pi$-base whose elements have an H-closed closure then such a space is Choquet (hence Baire).\\\noindent In Chapter \ref{Chapter2} we introduce some new selection principles involving networks, namely, M-{\it nw}-selective, R-{\it nw}-selective and H-{\it nw}-selective. We show that such spaces has countable fan tightness, countable strong fan tightness and the weak Fr\'echet in strict sense property, respectively, hence they are M-separable, R-separable and H-separable, respectively. Also they are Menger, Rothberger and Hurewicz.We give consistent results and we define \textit{trivial} R-, H-, and M-{\it nw}-selective spaces the ones with countable netweight 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. Moreover, using forcing methods, we construct consistent countable non-trivial examples of R-{\it nw}-selective and H-{\it nw}-selective spaces. Additionally, we establish some limitations to constructions of non-trivial examples and 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.\\\noindent In Chapter \ref{Chapter3} we introduce and investigate two new games called R-{\it nw}-selective game and the M-{\it nw}-selective game. These games naturally arise from the corresponding selection principles involving networks introduced by Bonanzinga and Giacopello.