I-weight, special base properties and related covering properties

Abstract

Alleche, Arhangel'ski\u{\i} and Calbrix defined the notion of a sharp base and posed the question: Is there a regular space with a sharp base whose product with [0, 1] does not have a sharp base? Chapter 2 contains an example of a space $P$ with a sharp base whose product with [0, 1] does not have a sharp base. The example in Chapter 2 also answers the following 3 questions found in the literature: Is every pseudocompact Tychonoff space with a sharp base metrizable? Is there a pseudocompact space $X$ with a G$_\delta$-diagonal and a point-countable base such that $X$ is not developable? Is every \v{C}ech-complete pseudocompact space with a point-countable base metrizable? The space we construct is pseudocompact, \v{C}ech-complete, has a G$_\delta$-diagonal, a sharp base and a point-countable base, but is not metrizable nor developable.

In Chapter 3, we study open-in-finite (OIF) bases and introduce the notion of a $\delta$-open-in-finite ($\delta$-OIF) base. Each $\delta$-OIF base is also OIF. We show that a base $\mathcal{B}$ for the space $X$ is $\delta$-OIF if and only if for each dense subset $Y$ of $X$, $\mathcal{B} \upharpoonright Y$ is OIF. We also define OIF-metacompact, $\delta$-OIF-metacompact, $(n, \kappa)$-metacompact, and $(<\omega, \kappa)$-metacompact and show that for generalized order spaces and $\kappa = \omega$ these properties are equivalent. The $(<\omega, \omega)$-metacompact property is corresponds to the $< \omega$-weakly uniform base property. We show that for Moore spaces $X$, the space $X$ has an OIF base (resp. $\delta$-OIF base, $<\omega$-weakly uniform base) if and only if the space is OIF-metacompact (resp. $\delta$-OIF-metacompact, $(<\omega, \omega)$-metacompact).

In the final chapter, we prove that for the class of linearly ordered compact spaces, i-weight reflects all cardinals. We find necessary and sufficient conditions for i-weight to reflect cardinal $\kappa$ in the class of locally compact linearly ordered spaces. In the last section we calculate the i-weight of paracompact spaces in terms of the local i-weight and extent of the space. This result determines that for compact spaces i-weight and local i-weight are the same.