Four metric conditions characterizing Čech dimension zero
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- by Kevin Broughan
- Proc. Amer. Math. Soc. 64 (1977), 176-178
- DOI: https://doi.org/10.1090/S0002-9939-1977-0515020-3
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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
- K. A. Broughan, A metric characterizing Čech dimension zero, Proc. Amer. Math. Soc. 39 (1973), 437–440. MR 314012, DOI 10.1090/S0002-9939-1973-0314012-3
- K. A. Broughan, Metrization of spaces having Čech dimension zero, Bull. Austral. Math. Soc. 9 (1973), 161–168. MR 341428, DOI 10.1017/S0004972700043082
- Kevin A. Broughan, Invariants for real-generated uniform topological and algebraic categories, Lecture Notes in Mathematics, Vol. 491, Springer-Verlag, Berlin-New York, 1975. MR 0425916, DOI 10.1007/BFb0080886
- R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR 0230273
Bibliographic Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 64 (1977), 176-178
- MSC: Primary 54E35; Secondary 54F45
- DOI: https://doi.org/10.1090/S0002-9939-1977-0515020-3
- MathSciNet review: 0515020