论文标题
开放着色公理
The Open Coloring Axiom
论文作者
论文摘要
这项工作与todorcěvić在\ cite {stevo}中引入的公理有关，该公理构成了关于真实拓扑的拉姆齐般的陈述。我们的目的是详细解释公理，提供一些有趣的应用，最后证明公理确实与ZFC一致，因此首先考虑使用它是有意义的。 对于这项特殊的学术努力，我们介绍了集合理论中的几个高级主题，包括{\ sl hausdorff gaps}，强迫，无限制组合和一点点拓扑。例如，我们采用基于Rothberger定理的参数来表明开放着色的公理意味着平等$ \ Mathfrak B = \ Aleph_2 $，这又使此公理与CH不一致。换句话说，在ZFC中，开放的着色公理可能是错误的。为了证明其相对的一致性，我们表明，通过遵循Todorcěvić的相当长而技术的引理，可以实现公理，这导致了这项工作的高潮。
This work is concerned with an axiom introduced by Todorcěvić in \cite{stevo} that constitutes a Ramsey-like statement regarding the topology of the reals. Our aim is to explain the axiom in detail, give some interesting applications and finally prove that the axiom is indeed consistent with ZFC, so that it makes sense to consider working with it in the first place. For this particular academic endeavor, we cover several advanced topics in set theory, including concepts like {\sl Hausdorff gaps}, forcing, infinitary combinatorics and a tad of topology. We employ, for example, an argument based on Rothberger's theorem to show that the Open Coloring Axiom implies the equality $\mathfrak b=\aleph_2$, which in turn makes this axiom inconsistent with CH. In other words, in ZFC, the Open Coloring Axiom could be false. To prove its relative consistency, we show that the axiom could be true by following a rather long and technical lemma of Todorcěvić, which leads to the culmination of this work.
