### Abstract

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 Hušek, 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.

Original language | English |
---|---|

Pages (from-to) | 610-623 |

Number of pages | 14 |

Journal | Topology and its Applications |

Volume | 221 |

DOIs | |

State | Published - Apr 15 2017 |

### Keywords

- Compact
- Compact Hausdorff space with cardinality ℵ
- Continuous images of the countable ordinals
- Countable ordinals
- Juhasz–Szentmiklossy theorem
- Locally compact
- Metrizable
- Monotonically normal
- Monotonically normal compactification
- Paracompact
- Scattered
- Small diagonals
- σ-Minimal base

## Fingerprint Dive into the research topics of 'Images of the countable ordinals'. Together they form a unique fingerprint.

## Cite this

*Topology and its Applications*,

*221*, 610-623. https://doi.org/10.1016/j.topol.2017.02.067