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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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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DOI: https://doi.org/10.1090/S0002-9939-1974-0334150-X
Keywords: Scattered space, derived length, weight, $ [1,\Omega )$
Article copyright: © Copyright 1974 American Mathematical Society