Combinatorics of Open Covers (I): Ramsey Theory

<p> We study several schemas for generating from one sort of open cover of a topological space a second sort of open cover. Some of these schemas come from classical literature, others are borrowed from the theory of ultra&filig;lters on the set of positive integers. We show that the fact that such a schema actually succeeds in producing a cover imposes strong combinatorial structure on the family of open covers of a certain sort. In particular, we show that certain analogues of Ramsey&rsquo;s theorem characterize some of these circumstances.</p>