Set-theoretic Structure of Compact Topological Spaces

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This research project concerns abstract mathematical spaces. In particular, we look to discover which types of such spaces can be said to exist (in the mathematical sense of the word). We also look at when these spaces can exist as sub-spaces of well-known spaces such as the common real line. As a precise example, if the pairs of real numbers are coloured with two colours, which kind of spaces will appear as sets for which all their pairs have the same colour? Many of the problems here are difficult in the sense that there are many statement about these spaces which can neither be proved nor refuted in the usual mathematical sense, with the usual mathematical assumptions. Moreover, we can prove that there can be no proof and we can also prove that there can be no refutation. This has serious implications for the study of the limitations on the strength of the mathematical method of inquiry.