On scattered spaces

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.