Set-theoretic structure of topological spaces

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Compact topological spaces are abstract structures much used in contemporary mathematical research. Inspired by geometry, but devoid of typically complex geometric structure, one could reasonably hope that their combinatorial structure would be simple and elementary. This is emphatically not the case. In fact there are many simply stated questions about compact topological spaces that are unresolved. Furthermore there is a growing number of such questions which cannot possibly be resolved today. That is, these statements can neither be logically proved nor logically disproved based upon commonly accepted mathematical principles. This research deals with such issues: in particular, the types of subspaces compact topological spaces must have. As well, when such spaces are partitioned into pieces, we investigate which sorts of subspaces must necessarily be included in at least one piece of the partition.