On scattered spaces

Kannan, V. ; Rajagopalan, M. (1974) On scattered spaces Proceedings of the American Mathematical Society, 43 (2). pp. 402-408. ISSN 0002-9939

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Official URL: http://dx.doi.org/10.1090/S0002-9939-1974-0334150-...

Related URL: http://dx.doi.org/10.1090/S0002-9939-1974-0334150-X

Abstract

We show that each O-dimensional Hausdorff space which is scattered can be mapped continuously in a one-to-one way onto a scattered O-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, Lindelof, scattered first-countable Hausdorff space admits a if it is scattered. We also show that a 0-dimensional, Lindelof, 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, Ω), and deduce a result of W. H. Young as a corollary.

Item Type:Article
Source:Copyright of this article belongs to American Mathematical Society.
Keywords:Scattered Space; Derived Length; Weight; [1, Ω)
ID Code:15549
Deposited On:13 Nov 2010 09:25
Last Modified:17 May 2016 00:26

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