Set-theoretic Structure of Compact Topological Spaces

这个研究项目涉及抽象数学空间。特别是，我们希望发现哪些类型的空间可以说是存在的（在数学意义上的话）。我们还研究了这些空间何时可以作为公知空间（如公共真实的直线）的子空间存在。作为一个精确的例子，如果真实的数对被涂上两种颜色，那么哪种空间将表现为所有对都具有相同颜色的集合？ 这里的许多问题是困难的，在这个意义上，有许多声明，这些空间既不能证明，也不能反驳在通常的数学意义上，与通常的数学假设。此外，我们可以证明没有证据，我们也可以证明没有反驳。这对于研究数学探究方法的局限性具有重要意义。

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.