Set-theoretic structure of topological spaces

紧致拓扑空间是现代数学研究中常用的抽象结构。受到几何的启发，但缺乏典型的复杂几何结构，人们可以合理地希望它们的组合结构将是简单和基本的。这显然不是事实。事实上，有许多关于紧致拓扑空间的简单问题是没有解决的。此外，今天不可能解决的这类问题越来越多。也就是说，根据普遍接受的数学原理，这些陈述既不能被逻辑证明，也不能被逻辑证伪。本文主要研究紧拓扑空间必须具有的子空间类型。同样，当这样的空间被分割成块时，我们研究了哪些种类的子空间必须包含在分割的至少一个块中。

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.