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Four metric conditions characterizing Čech dimension zero

Author: Kevin Broughan
Journal: Proc. Amer. Math. Soc. 64 (1977), 176-178
MSC: Primary 54E35; Secondary 54F45
MathSciNet review: 0515020
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Abstract: If (X,d) is a metric space let $ {d_x}(y) = d(x,y)$. It is proved that if each x in X has a neighbourhood P with $ {d_x}(P)$ not dense in any neighbourhood of 0 in $ [0,\infty )$ then Ind $ X = 0$. This metric condition characterizes metrizable spaces which have Čech dimension zero. Three other metric characterizations are given.

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

  • [1] K. A. Broughan, A metric characterizing Čech dimension zero, Proc. Amer. Math. Soc. 39 (1973), 437-440. MR 0314012 (47:2564)
  • [2] -, Metrization of spaces having Čech dimension zero, Bull. Austral. Math. Soc. 9 (1973), 161-168. MR 0341428 (49:6179)
  • [3] -, Invariants for real-generated uniform topological and algebraic categories, Lecture Notes in Math., vol. 491, Springer-Verlag, Berlin and New York, 1975. MR 0425916 (54:13866)
  • [4] R. Engelking, Outline of general topology, North-Holland, Amsterdam, 1968. MR 0230273 (37:5836)

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Keywords: Metric spaces, Čech dimension zero
Article copyright: © Copyright 1977 American Mathematical Society

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