Discrete sets, free sequences and cardinal properties of topological spaces

Abstract

We study the influence of discrete sets and free sequences on cardinal properties of topological spaces. We focus mainly on the minimum number of discrete sets needed to cover a space X (denoted by dis(X)) and on reflection of cardinality by discrete sets, free sequences and their closures. In particular, we offer several classes of spaces such that the minimum number of discrete sets required to cover them is always bounded below by the "dispersion character" (i.e., minimum cardinality of a non-empty open set). Two of them are Baire generalized metric spaces, and the rest are classes of compacta. These latter classes offer several partial positive answers to a question of Juhasz and Szentmiklossy. In some cases we can weaken compactness to the Baire property plus some other good property. However, we construct a Baire hereditarily paracompact linearly ordered topological space such that the gap between dis(X) and the dispersion character can be made arbitrarily big. We show that our results about generalized metric spaces are sharp by constructing examples of good Baire generalized metric spaces whose dispersion character exceeds the minimum number of discrete sets required to cover them. With regard to discrete reflection of cardinality we offer a series of improvements to results of Alan Dow and Ofelia Alas. We introduce a rather weak cardinal function, the "breadth", defined as the supremum of cardinalities of closures of free sequences in a space, and prove some instances where it manages to reflect cardinality. We finish with a common generalization of Arhangel'skii Theorem and De Groot's inequality and its increasing chain version.