Set-theoretic structure of topological spaces

紧拓扑空间是现代数学研究中常用的抽象结构。受几何学的启发，但没有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是简单和基本的。事实显然并非如此。事实上，关于紧致拓扑空间有许多简单明了的问题尚未解决。此外，这类问题越来越多，今天不可能解决。也就是说，这些陈述既不能在逻辑上证明，也不能根据普遍接受的数学原理在逻辑上证明。本文主要研究了紧拓扑空间的子空间类型。同样，当这样的空间被划分成块时，我们研究哪些类型的子空间必须包含在至少一个分区中。

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.