Mathematical Sciences: Problem in Set-Theoretic Topology

9322613 nykos这个项目的目标是一些问题领域，在这些领域，拓扑学家和集合理论家之间的相互作用似乎有可能在这两个领域都取得重大进展。由于这种相互作用，集合论拓扑学在过去二十年中得到了爆炸性的发展，也激发了集合论领域许多卓有成效的研究。然而，在过去的几年中，结果流显示出减弱的迹象，并且可能需要开发新的集合论公理和技术来保持拓扑领域的良好健康。特别强调的问题领域是遗传的常态；可数紧空间的结构与积以及不可度量流形的理论。集合论拓扑学是一门比其他拓扑学更基础的学科，主要涉及非常抽象的空间，以及集合论公理变体的采用对拓扑学的影响。其次，它不仅可以应用于集合论和其他拓扑，还可以应用于分析和代数，特别是布尔代数理论。Nyikos的研究强调了所有这些方面。例如，他对可以施加在长线上的微分结构（即平滑）及其切向量的相关二维流形进行了广泛的研究，为那些希望理解更熟悉但更高维流形的非平凡平滑的人提供了有价值的启发。（这在今天是一个非常重要的课题，尤其是在与理论物理相互作用很强的四维空间中。）长直线的一些平滑需要在集合论的通常公理之外的公理。Nyikos是在集合论拓扑中使用这些公理的领导者，因为他在十多年前发现了一个关于摩尔空间的著名的未被证明的猜想实际上独立于集合论的通常基本公理。此外，这个猜想被证明与康托连续统假设不一致，康托连续统假设是一个更熟悉的命题，大多数工作的数学家如果必须做出选择，他们可能会选择康托连续统假设。这一结果表明了最佳集合论拓扑的特点。* * *

9322613 Nyikos This project is aimed at some problem areas where interaction between topologists and set theorists seems likely to lead to significant progress in both areas. Set-theoretic topology has grown explosively in the last two decades as a result of such interaction, stimulating much fruitful research in set theory as well. In the last few years, however, the flow of results has shown signs of abating, and the development of new set-theoretic axioms and techniques will probably be needed to keep this area of topology in good health. The problem areas on which particular stress will be laid are hereditary normality; the structure and products of countably compact spaces; and the theory of nonmetrizable manifolds. Set-theoretic topology is more of a foundational subject than the rest of topology, primarily concerning itself with very abstract spaces and with the impact on topology of the adoption of variants of the axioms of set theory. Secondarily, it has applications not only to set theory and the rest of topology, but also to analysis and algebra, particularly the theory of Boolean algebras. Nyikos's research has emphasized all these aspects. For example, he has done an extensive study of the differential structures (i.e. smoothings) that can be imposed on the long line, and of the associated 2-dimensional manifold of its tangent vectors, providing valuable heuristics for those wishing to understand nontrivial smoothings of more familiar but higher-dimensional manifolds. (This is a subject of major importance today, especially in 4 dimensions, where the interplay with theoretical physics is strong.) Some of the smoothings of the long line required axioms outside the usual axioms of set theory. Nyikos has been a leader in the use of these axioms in set-theoretic topology ever since his involvement over a decade ago in the discovery that a famous unproved conjecture concerning Moore spaces was actually independent of the usual basic axiom s of set theory. Moreover, this conjecture was shown to be inconsistent with the Cantor Continuum Hypothesis, a far more familiar proposition that most working mathematicians would be likely to choose if they had to make a choice. This result suggests the flavor of the best set-theoretic topology. ***