I-weight, special base properties and related covering properties.

摘要

Alleche, Arhangel'skii˘ 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 Gdelta-diagonal and a point-countable base such that X is not developable? Is every Cech-complete pseudocompact space with a point-countable base metrizable? The space we construct is pseudocompact, Cech-complete, has a Gdelta-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 B for the space X is delta-OIF if and only if for each dense subset Y of X, B↾Y is OIF. We also define OIF-metacompact, delta-OIF-metacompact, ( n, k)-metacompact, and ( o, kappa)-metacompact and show that for generalized order spaces and kappa = o these properties are equivalent. The ( o, o)-metacompact property is corresponds to the o-weakly uniform base property. We show that for Moore spaces X, the space X has an OIF base (resp. delta-OIF base, o-weakly uniform base) if and only if the space is OIF-metacompact (resp. delta-OIF-metacompact, ( o, o)-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.