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On scattered spaces

Authors: V. Kannan and M. Rajagopalan
Journal: Proc. Amer. Math. Soc. 43 (1974), 402-408
MSC: Primary 54D20; Secondary 54A25
MathSciNet review: 0334150
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Abstract: We show that each 0-dimensional Hausdorff space which is scattered can be mapped continuously in a one-to-one way onto a scattered 0-dimensional Hausdorff space of the same weight as its cardinality. This gives an easier and a new proof of the fact that a countable regular space admits a coarser compact Hausdorff topology if and only if it is scattered. We also show that a 0-dimensional, Lindelöf, scattered first-countable Hausdorff space admits a scattered compactification. In particular we give a more direct proof than that of Knaster, Urbanik and Belnov of the fact that a countable scattered metric space is a subspace of $ [1,\Omega )$, and deduce a result of W. H. Young as a corollary.

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  • [1] V. K. Bel'nov, On condensations onto compacta, Dokl. Akad. Nauk SSSR 193 (1970), 506-509=Soviet Math. Dokl. 11 (1970), 949-952. MR 42 #3756. MR 0268859 (42:3756)
  • [2] V. Kannan and M. Rajagopalan, Countable topological spaces, Technical Publications of the Department of Mathematics, Madurai University (to appear).
  • [3] M. Katetov, On the mappings of countable spaces, Colloq. Math. 2 (1949), 30-33. MR 12, 627. MR 0039988 (12:627a)
  • [4] B. Knaster and K. Urbanik, Sur les espaces complets séparables de dimension 0, Fund. Math. 40 (1953), 194-202. MR 15, 641. MR 0060221 (15:641h)
  • [5] S. Mazurkiewicz and W. Sierpinski, Contribution à la topologies des ensembles d'enombrables, Fund. Math. 1 (1920), 17-27.
  • [6] S. Mrowka, M. Rajagopalan and T. Soundararajan, A characterisation of compact scattered spaces through chain limits (Chain compact spaces), Proc. Pittsburgh Conference in Topology II (1972), Academic Press, New York (to appear). MR 0375234 (51:11430)
  • [7] R. S. Pierce, Existence and uniqueness theorems for extensions of zero-dimensional compact metric spaces, Trans. Amer. Math. Soc. 148 (1970), 1-21. MR 40 #8011. MR 0254804 (40:8011)
  • [8] C. Ryll-Nardzewski and R. Telgarsky, On scattered compactification, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 233-234. MR 41 #7635. MR 0263030 (41:7635)
  • [9] Z. Semadeni, Sur les ensembles clairsemés, Rozprawy Math. 19 (1959), 39 pp. MR 21 #6571. MR 0107849 (21:6571)
  • [10] W. Sierpinski, Sur une propriété topologique des ensembles denombrables denses en soi, Fund. Math. 1 (1920), 11-16.

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Keywords: Scattered space, derived length, weight, $ [1,\Omega )$
Article copyright: © Copyright 1974 American Mathematical Society

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