| Popis: |
We study spaces that are continuous images of the usual space [ 0 , ω 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 δ -diagonal, Y is perfect, Y has a point-countable base, Y has a small diagonal in the sense of Husek, and Y has a σ-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 | Y | ≤ ℵ 1 and having a small diagonal, then Y is metrizable, and we deduce a recent result of Gruenhage from work of Mrowka, Rajagopalan, and Soundararajan. |
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