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A generalization of the Sierpiński theorem

Author: J. J. Dijkstra
Journal: Proc. Amer. Math. Soc. 91 (1984), 143-146
MSC: Primary 54F60
MathSciNet review: 735581
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Abstract: Sierpiński's theorem admits the following generalization. Let $ n$ be a nonnegative integer and $ X$ a compact Hausdorff space. If $ \left\{ {{F_i}\left\vert {i \in {\mathbf{N}}} \right.} \right\}$ is a countable closed covering of $ X$ such that $ \left( {{F_i} \cap {F_j}} \right) < n$ for distinct $ i$ and $ j$ in $ {\mathbf{N}}$, then every continuous mapping from $ {F_1}$ into the $ n$-sphere $ {S^n}$ is extendable over $ X$.

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

  • [1] J. J. Dijkstra, Fake topological Hilbert spaces and characterizations of dimension in terms of negligibility, CWI Tract, vol. 2, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. MR 753464
  • [2] Ryszard Engelking, Dimension theory, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author; North-Holland Mathematical Library, 19. MR 0482697
  • [3] -, General topology, PWN, Warsaw, 1977.
  • [4] W. Sierpiński, Un théorème sur les continus, Tôhoku Math. J. 13 (1918), 300-303.

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Keywords: Sierpiński's theorem, covering dimension, $ n$-sphere
Article copyright: © Copyright 1984 American Mathematical Society