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Metrizable spaces where the inductive dimensions disagree


Author: John Kulesza
Journal: Trans. Amer. Math. Soc. 318 (1990), 763-781
MSC: Primary 54F45
DOI: https://doi.org/10.1090/S0002-9947-1990-0954600-9
MathSciNet review: 954600
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Abstract: A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roy's technique, these spaces can often be shown to have positive large inductive dimension. Examples of $ {\mathbf{N}}$-compact, complete metrizable spaces with $ \operatorname{ind} = 0$ and $ \operatorname{Ind} = 1$ are provided, answering questions of Mrowka and Roy. An example with weight $ \mathfrak{c}$ and positive Ind such that subspaces with smaller weight have $ \operatorname{Ind} = 0$ is produced in ZFC. Assuming an additional axiom, for each cardinal $ \lambda $ a space of positive Ind with all subspaces with weight less than $ \lambda $ strongly zero-dimensional is constructed.


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  • [A] C. E. Aull, Problem section, Lecture Notes in Pure and Appl. Math., vol. 95, Marcel Dekker, New York and Basel, 1985, pp. 311-314. MR 789281 (86g:54019)
  • [F] W. G. Fleissner, The normal Moore space conjecture and large cardinals, Handbook of Set Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 733-760. MR 776635 (86m:54023)
  • [K1] J. Kulesza, An example in the dimension theory of metrizable spaces, Topology Appl. (to appear). MR 1058791 (91g:54045)
  • [K2] -, Dissertation, SUNY at Binghamton, 1987.
  • [M] S. Mrowka, $ {\mathbf{N}}$-compactness, metrizability, and covering dimension, Lecture Notes in Pure and Appl. Math., vol. 95, Marcel Dekker, New York and Basel, 1985, pp. 247-275. MR 789276 (86i:54034)
  • [N] G. Naber, Set theoretic topology, University Microfilms International, Ann Arbor, Mich., 1977.
  • [NA] K. Nagami, Dimension theory, Academic Press, New York, 1970. MR 0271918 (42:6799)
  • [NY] P. J. Nyikos, Prabir Roy's space $ \Delta $ is not $ {\mathbf{N}}$-compact, General Topology Appl. 3 (1973), 197-210. MR 0324657 (48:3007)
  • [O] A. Ostaszewski, A note on the Prabir Roy space $ \Delta $, preprint.
  • [R1] P. Roy, Failure of equivalence of dimension concepts for metric spaces, Bull. Amer. Math. Soc. 68 (1962), 609-613. MR 0142102 (25:5495)
  • [R2] -, Nonequality of dimensions for metric spaces, Trans. Amer. Math. Soc. 134 (1968), 117-132. MR 0227960 (37:3544)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0954600-9
Keywords: Metrizable space, full set, dimension
Article copyright: © Copyright 1990 American Mathematical Society

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