具有σ-点有限基的空间

General Topology and its Applications Pub Date : 1978-04-01 DOI:10.1016/0016-660X(78)90003-X

W.N. Hunsaker, W.F. Lindgren

引用次数: 0

摘要

定理。设X是T1空间。下列是等价的:(1)X有一个σ-不相交的基。(FX2)X是拟可展开的并且有一个基是秩1集合序列的并。(3)X有一个拟可展开(Gn)，其性质是:对于每一个X， {st2(X,Gn): X∈st2(X,Gn)， n是n (X)的一个基。定理。设X是T1空间。以下是等价的:(1)X有一个σ-点有限基(2)X对每一个Gn都有一个准展开(Gn)， (3)X对每一个次无限秩的Gn noetheran都有一个准展开(Gn)， (4)X对每一个有限秩的Gn noetheran都有一个准展开(Gn)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Spaces with σ-point finite bases

Theorem. Let X be a T₁ space. The following are equivalent:

(1)
X has a σ-disjoint base.
(FX2)
X is quasi-developable and has a base that is the union of a sequence of rank 1 collections.
(3)
X has a quasi-development ( $G$ _n) with the property that for each x, ${st^{2} (x, G_{n}): x ∈st^{2} (x, G_{n})$ , n a positive integer} is a base for $N$ (x).

Theorem. Let X be a T₁ space. The following are equivalent:

(1)
X has a σ-point finite base.
(2)
X has a quasi-development ( $G$ _n) with each $G$ _n well ranked.
(3)
X has a quasi-development ( $G$ _n) with each $G$ _n Noetherian of sub-infinite rank.
(4)
X has a quasi-development ( $G$ _n) with each $G$ _n Noetherian of point finite rank.

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General Topology and its Applications

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