Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological properties of Souslin subsets

Author: R. W. Hansell
Journal: Trans. Amer. Math. Soc. 293 (1986), 613-622
MSC: Primary 54H05; Secondary 04A15, 54D15
MathSciNet review: 816314
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a subparacompact regular space such that the projection map $ p:X \times {\mathbf{P}} \to X$, where $ {\mathbf{P}}$ is the space of irrational numbers, preserves collections of sets having a $ \sigma $-locally finite refinement. It is shown that $ p$ then preserves generalized $ {F_\sigma }$-sets. It follows that, if $ X$ has any tpological property which is hereditary with respect to generalized $ {F_\sigma }$-sets, then every Souslin subset of $ X$ will also have this property in the relative topology. Such topological properties include nearly all covering properties (paracompactness, metacompactness, etc.), as well as normality, collectionwise normality, and the Lindelöf property. We show that the above mapping property will hold whenever $ X$ is a $ P(\omega )$-space, thus, in particular, when $ X$ is any Souslin (hence any Baire) subset of a compact space crossed with a metrizable space. Additional topological properties of Souslin subsets, such as topological completeness, realcompactness, $ z$-embeddedness, and the properties of being a $ P$-space or $ \Sigma $-space, are also considered.

References [Enhancements On Off] (What's this?)

  • [1] R. A. Alo and H. L. Shapiro, Normal topological spaces, Cambridge Univ. Press, Cambridge, 1974. MR 0390985 (52:11808)
  • [2] D. Burke, Covering properties, Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984. MR 776628 (86e:54030)
  • [3] W. W. Comfort and S. Negrepontis, Continuous pseudometrics, Lecture Notes in Pure and Appl. Math., Vol. 14, Dekker, New York, 1975. MR 0410618 (53:14366)
  • [4] R. Engelking, General Tpology, Polish Scientific Publishers, Warsaw, 1977. MR 0500780 (58:18316b)
  • [5] R. W. Hansell, J. E. Jayne and C. A. Rogers, $ K$-analytic sets, Mathematika 30 (1983), 189-221. MR 737176 (85b:54059)
  • [6] R. W. Hansell, J. E. Jayne and C. A. Rogers, Separation of $ K$-analytic sets, Mathematika (to appear). MR 737176 (85b:54059)
  • [7] J. E. Jayne and C. A. Rogers, $ K$-analytic sets, Analytic Sets, Academic Press, London, 1980.
  • [8] D. J. Lutzer, Another property of the Sorgenfrey line, Compositio Math. 24 (1972), 359-363. MR 0307171 (46:6292)
  • [9] E. Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. 4 (1953), 831-838. MR 0056905 (15:144b)
  • [10] -, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963), 375-376. MR 0152985 (27:2956)
  • [11] -, On Nagami's $ \Sigma $-spaces and some related matters, Proc. Washington State Univ. Conf. on General Topology, Pullman, Wash., 1970, pp. 13-19.
  • [12] -, On maps related to $ \sigma $-locally finite and $ \sigma $-discrete collections of sets, Pacific J. Math. 98 (1982), 139-152. MR 644945 (83b:54012)
  • [13] K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365-382. MR 0165491 (29:2773)
  • [14] K. Nagami, $ \Sigma $-spaces, Fund. Math. 65 (1969), 169-192. MR 0257963 (41:2612)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H05, 04A15, 54D15

Retrieve articles in all journals with MSC: 54H05, 04A15, 54D15

Additional Information

Keywords: Souslin set, generalized $ {F_\sigma }$-set, covering properties, $ \Sigma $-space, $ P$-space, refinement $ \sigma $-locally finite maps
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society