### Abstract

The following result due to Hanai, Morita, and Stone is well known: Let f be a closed continuous map of a metric space X onto a topological space Y. Then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) for each y ∈ Y, f^{- 1} {y} has a compact boundary in X; (iii) Y is metrizable. In this article we obtain several related results in the setting of topological ordered spaces. In particular we investigate the upper and lower topologies of metrizable topological ordered spaces which are both C- and I-spaces in the sense of Priestley.

Original language | English |
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Pages (from-to) | 2914-2922 |

Number of pages | 9 |

Journal | Topology and its Applications |

Volume | 156 |

Issue number | 18 |

DOIs | |

Publication status | Published - Dec 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*156*(18), 2914-2922. https://doi.org/10.1016/j.topol.2008.12.040

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*Topology and its Applications*, vol. 156, no. 18, pp. 2914-2922. https://doi.org/10.1016/j.topol.2008.12.040

**Topological ordered C- (resp. I-)spaces and generalized metric spaces.** / Künzi, Hans Peter A.; Mushaandja, Zechariah.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Topological ordered C- (resp. I-)spaces and generalized metric spaces

AU - Künzi, Hans Peter A.

AU - Mushaandja, Zechariah

PY - 2009/12/1

Y1 - 2009/12/1

N2 - The following result due to Hanai, Morita, and Stone is well known: Let f be a closed continuous map of a metric space X onto a topological space Y. Then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) for each y ∈ Y, f- 1 {y} has a compact boundary in X; (iii) Y is metrizable. In this article we obtain several related results in the setting of topological ordered spaces. In particular we investigate the upper and lower topologies of metrizable topological ordered spaces which are both C- and I-spaces in the sense of Priestley.

AB - The following result due to Hanai, Morita, and Stone is well known: Let f be a closed continuous map of a metric space X onto a topological space Y. Then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) for each y ∈ Y, f- 1 {y} has a compact boundary in X; (iii) Y is metrizable. In this article we obtain several related results in the setting of topological ordered spaces. In particular we investigate the upper and lower topologies of metrizable topological ordered spaces which are both C- and I-spaces in the sense of Priestley.

UR - http://www.scopus.com/inward/record.url?scp=70349466203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349466203&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2008.12.040

DO - 10.1016/j.topol.2008.12.040

M3 - Article

AN - SCOPUS:70349466203

VL - 156

SP - 2914

EP - 2922

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 18

ER -