Images of the countable ordinals

We study spaces that are continuous images of the usual space [0, omega(1)) of countable ordinals. We begin by showing that if Y is such a space and is T-3 then Y has a monotonically normal compactification, and is monotonically normal, locally compact and scattered. Examples show that regularity is needed in these results. We investigate when a regular continuous image of the countable ordinals must be compact, paracompact, and metrizable. For example we show that metrizability of such a Y is equivalent to each of the following: Y has a G(delta)-diagonal, Y is perfect, Y has a point-countable base, Y has a small diagonal in the sense of Husek, and Y has a sigma-minimal base. Along the way we obtain an absolute version of the Juhasz-Szentmiklossy theorem for small spaces, proving that if Y is any compact Hausdorff space having vertical bar Y vertical bar <= N-1 Iti and having a small diagonal, then Y is metrizable, and we deduce a recent result of Gruenhage from work of Mrowka, Rajagopalan, and Soundararajan. (C) 2017 Elsevier B.V. All rights reserved.