Countable Convergence in Compact Spaces

PI研究紧空间类中可数收敛问题的独立性结果。即使在适当的原力公理存在的情况下，一些问题仍然没有得到解决。空间的序数是一个衡量序列空间复杂性的自然序数不变量；紧空间的这个不变量上是否有一个有限或可数的界的问题只在CH下得到了解决。我们还将研究有限到一的整数的Stone-Cech紧化的剩余部分上的连续映射。我们将试图确定这样的映射是否必须是“某处微不足道的”，以及值域空间是否与域同胚。这个问题在PFA下已经完全解决了，但例如在MA下还没有得到解决。关于PFA下N的剩余的另外两个问题：它是否可以被无处稠密的P-集合所覆盖？是否存在剩余部分的非平凡副本？当人们形式化无穷步序列的极限过程时，自然会引出拓扑空间的概念。物理空间中的点可以使用收敛序列的概念进行分析，但正如数学需要对点有更广泛的理解(例如，函数或过程本身位于类似过程的空间中)一样，对极限和拓扑空间的更广泛理解也需要在更一般的环境中发展为手工收敛。对广义收敛的本质、可能隐含(更容易理解和应用的)经典收敛序列的条件之间的联系以及在变形下收敛结构的行为的研究是持续和富有成效的。这些概念对数学的基本公理非常敏感，这些问题既是基础研究的试验台，也是新发展的生成器。连续体假设(实数线的“大小”)是数学中最著名的不确定问题之一，它与这些收敛问题密切相关。

The PI investigates independence results in the study of countableconvergence questions in the class of compact spaces. Some of thequestions remain unresolved even in the presence of the Proper ForcingAxiom. The sequential order of a space is a natural ordinal invariantthat measures the complexity of a sequential space; the question ofwhether there can be a finite or countable bound on this invariant forcompact spaces has only been resolved under CH. We will also studycontinuous maps on the remainder of the Stone-Cech compactification ofthe integers which are finite-to-one. We will try to determine if suchmaps have to be "somewhere trivial" and whether the range space will behomeomorphic to the domain. This question has been fully resolved underPFA but not, for example, under MA. Two other questions about theremainder of N under PFA: can it be covered by nowhere dense P-sets andcan there be non-trivial copies of the remainder?When one formalizes the limiting process of an infinite sequence ofsteps one is naturally lead to the notion of topological space. Pointsin physical space can be analyzed using the notion of a convergingsequence, but just as mathematics has a need for a much broaderunderstanding of points (e.g. functions or processes themselves lying ina space of like processes), so too has the need for a broaderunderstanding of limit and topological space been developed to handleconvergence in the more general settings. The investigation into thenature of generalized convergence, connections between conditions thatmay imply the existence of (the much easier to understand and apply)classical convergent sequences, and the behavior of the convergencestructure under deformations is on-going and productive. These notionsare very sensitive to the foundational axioms of mathematics and thequestions serve as both a testbed for foundational study and a generatorof new development. One of the most famously indeterminate questions ofmathematics, the continuum hypothesis (the "size" of the real numberline) is intimately connected to these questions of convergence.