### Abstract

A space is weakly θ-refinable if every open cover U of X has an open refinement V = ∪{V(n): n≥ 1} such that given xε{lunate}X, one of the collections V(n) has finite, positive order at x. Several equivalent properties of a space are given and are used to prove that: (a) if X is weakly θ-refinable and has closed sets G_{δ} then X is subparacompact; (b) any quasi-developable space (in the sense of Bennett) is weekly θ-refinable; (c) a space is quasi-developable if and only if it has a θ-base, (d) a linearly ordered topological space is paracompact if and only if it is weakly θ-refinable. Examples are given which show that weak θ-refinability is strictly weaker than the notion of θ-refinability introduced by Worrell and Wicke.

Original language | English |
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Pages (from-to) | 49-54 |

Number of pages | 6 |

Journal | General Topology and its Applications |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - May 1972 |

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### Cite this

*General Topology and its Applications*,

*2*(1), 49-54. https://doi.org/10.1016/0016-660X(72)90035-9