Cardinal invariants and covering properties in topology
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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.