Countable Convergence in Compact Spaces

PI研究了紧空间类中可数收敛问题的独立性结果。有些问题即使在真力公理存在的情况下仍然没有得到解决。空间的序是一个自然的序不变量，它度量了序空间的复杂性;紧空间的这个不变量是否有有限或可数的界的问题只有在CH下才得到解决。我们还将研究整数的Stone-Cech紧化的剩余部分上的连续映射，这些整数是有限到1的。我们将试图确定这样的映射是否必须是“某个平凡的地方”，以及值域空间是否同胚于定义域。这一问题在《行动纲领》下已得到充分解决，但在《千年评估》等文书下却没有得到解决。关于PFA下N的余项的另外两个问题：它能被无处稠密的P-集覆盖吗？当一个人形式化的限制过程的一个无限序列的步骤之一是自然导致的概念拓扑空间。点的物理空间可以用收敛序列的概念来分析，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，也需要对极限和拓扑空间有更广泛的理解，以便在更一般的设置中处理收敛。对广义收敛的性质、可能暗示经典收敛序列存在的条件（更容易理解和应用）之间的联系以及变形下收敛结构的行为的研究正在进行中并且富有成效。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。数学中最著名的不确定问题之一，连续统假设（真实的数线的“大小”）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.