Set-theoretic structure of topological spaces

紧拓扑空间是当代数学研究中广泛使用的抽象结构。受到几何学的启发，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初级的。情况肯定不是这样的。事实上，有许多关于紧拓扑空间的简单陈述的问题尚未解决。此外，越来越多的此类问题今天不可能得到解决。也就是说，根据公认的数学原理，这些陈述既不能在逻辑上证明，也不能在逻辑上反驳。本文主要研究了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片段时，我们调查哪些类型子空间必须包括在至少一个分割片段中。

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.