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A study of γ-metrizable spaces
Topology and its Applications ( IF 0.6 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.topol.2020.107327
Liang-Xue Peng , Ying-Kun Xu

Abstract Let γ be an open cover of a space X. If there is a metric d on X which metrizes each member of γ, then X is said to be γ-metrizable. This notion was introduced and studied by M.A. Al Shumrani in [20] . We give an example to show that there is a gap in the proof of Theorem 2.1 and we give an example of a regular γ-metrizable space which is not metrizable, showing that main Theorems 2.1 and 2.4 in Al Shumrani's article are false. We discuss some properties of γ-metrizable spaces and get the following conclusions. If X n is a γ n -metrizable space for every n ∈ N , then the product space X = ∏ n ∈ N X n is γ-metrizable for the natural open cover γ of X. If X has a point-finite open cover by semi-stratifiable subspaces of X, then X is semi-stratifiable. Hence X is a semi-stratifiable space if X is a metacompact γ-metrizable space. If X is a regular γ-metrizable space and the family γ is a σ-HCP open cover of X, then X is metrizable. A space X is metrizable if and only if X is a paracompact γ-metrizable space. Every T 1 -space X with a σ-weakly hereditarily closure-preserving network is a D-space. We finally get that if X is a γ-metrizable space and the family γ is a σ-weakly hereditarily closure-preserving open cover of X, then X is a D-space.

中文翻译:

γ-可度量空间的研究

摘要 令γ 是空间X 的开覆盖。如果X 上存在度量d 来度量γ 的每个成员,则称X 是γ-可度量的。这个概念是由 MA Al Shumrani 在 [20] 中引入和研究的。我们举一个例子来证明定理2.1的证明存在差距,我们举了一个不可度量化的规则γ-可度量空间的例子,表明Al Shumrani的文章中的主要定理2.1和2.4是错误的。我们讨论了 γ 可度量空间的一些性质并得到以下结论。如果 X n 是每个 n ∈ N 的 γ n 可度量空间,则乘积空间 X = ∏ n ∈ NX n 是 X 的自然开覆盖 γ 的 γ 可度量化空间。如果 X 有一个点有限开覆盖X 的半可分层子空间,则 X 是半可分层的。因此,如果 X 是元紧致的 γ 可度量空间,则 X 是半分层空间。如果 X 是正则 γ-可度量空间并且族 γ 是 X 的 σ-HCP 开覆盖,则 X 是可度量的。空间 X 是可度量的当且仅当 X 是一个超紧 γ 可度量空间。每个 T 1 -空间 X 具有 σ - 弱遗传性闭包保留网络是一个 D 空间。我们最终得到,如果 X 是一个 γ-可度量空间,而家庭 γ 是 X 的一个 σ-弱遗传性保持闭包的开覆盖,那么 X 是一个 D-空间。
更新日期:2020-08-01
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