Countable Convergence in Compact Spaces

PI研究紧空间中可数收敛问题的独立性结果。有些问题即使在适当强制公理的存在下仍然没有解决。空间的有序阶是一个自然有序不变量，用来衡量一个有序空间的复杂度；对于紧空间的这个不变量是否存在有限界或可数界的问题，只在CH下得到了解决。我们还将研究有限到1整数的stone - ech紧化余数上的连续映射。我们将尝试确定这样的映射是否必须是“平凡的某处”，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，但在MA下却没有。关于PFA下N的余数的另外两个问题：它能被无处可寻的密集p集覆盖吗？余数能有非平凡的副本吗？当一个人形式化无穷步长序列的极限过程时，就会自然而然地得到拓扑空间的概念。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，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，可能暗示存在（更容易理解和应用）经典收敛序列的条件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，也是新发展的源泉。连续统假设（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.