Cardinal invariants and covering properties in topology

Covering properties are among the most important properties in topology. In most one wants to show certain covers can be reduced to subcovers of lower cardinality. A well-studied property, of which many of these are special cases, is ([alpha],[beta]) -compact which means any open cover of cardinality at most [beta] contains a subcover of cardinality less than [alpha]. We introduce two cardinally dependent properties which we call cocompact and chain cocompact since they involve complements of cover elements and are closely related to ([alpha],[beta]) -compact. In fact, for appropriate cardinals, cocompact implies compact which in turn implies chain cocompact. We also define a cardinally dependent version of locally compact which has implications about known cardinal invariants. Topologists have derived many results for a space which is a union of a chain (ordered by inclusion) of spaces having a certain property. Using this technique we derive some results involving initial compactness and examine under what conditions a k-bounded space can be initially k[superscript]+-compact and even k[superscript]+-bounded. Finally, we examine the structure of [omega]-bounded manifolds and how [omega]-cocompactness applies to it