On scattered spaces

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.