### 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 (US) |
---|---|

Pages (from-to) | 610-623 |

Number of pages | 14 |

Journal | Topology and its Applications |

Volume | 221 |

DOIs | |

State | Published - Apr 15 2017 |

### Fingerprint

### 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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*221*, 610-623. DOI: 10.1016/j.topol.2017.02.067

**Images of the countable ordinals.** / Bennett, Harold; Davis, Sheldon; Lutzer, David.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol 221, pp. 610-623. DOI: 10.1016/j.topol.2017.02.067

}

TY - JOUR

T1 - Images of the countable ordinals

AU - Bennett,Harold

AU - Davis,Sheldon

AU - Lutzer,David

PY - 2017/4/15

Y1 - 2017/4/15

N2 - We study spaces that are continuous images of the usual space [0,ω1) of countable ordinals. We begin by showing that if Y is such a space and is T3 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 Gδ-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|≤ℵ1 and having a small diagonal, then Y is metrizable, and we deduce a recent result of Gruenhage from work of Mrowka, Rajagopalan, and Soundararajan.

AB - We study spaces that are continuous images of the usual space [0,ω1) of countable ordinals. We begin by showing that if Y is such a space and is T3 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 Gδ-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|≤ℵ1 and having a small diagonal, then Y is metrizable, and we deduce a recent result of Gruenhage from work of Mrowka, Rajagopalan, and Soundararajan.

KW - Compact

KW - Compact Hausdorff space with cardinality ℵ

KW - Continuous images of the countable ordinals

KW - Countable ordinals

KW - Juhasz–Szentmiklossy theorem

KW - Locally compact

KW - Metrizable

KW - Monotonically normal

KW - Monotonically normal compactification

KW - Paracompact

KW - Scattered

KW - Small diagonals

KW - σ-Minimal base

UR - http://www.scopus.com/inward/record.url?scp=85014004353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014004353&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2017.02.067

DO - 10.1016/j.topol.2017.02.067

M3 - Article

VL - 221

SP - 610

EP - 623

JO - Topology and its Applications

T2 - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -