We continue Gartside, Moody, and Stares' study of versions of monotone paracompactness. We show that the class of spaces with a monotone closure-preserving open operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open locally-finite operator, and so do GO-spaces with a monotone (open or not) closure-preserving operator, whose underlying LOTS has a $\sigma$-closed-discrete dense subset. A GO-space with a $\sigma$-closed-discrete dense subset and a monotone closure-preserving operator is metrizable. A compact LOTS with a monotone open closure-preserving operator is metrizable.

中文翻译：

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由闭包保留运算符定义的单调覆盖属性

我们继续 Gartside、Moody 和 Stares 对单调超紧性版本的研究。我们表明，具有单调闭包保持开算符的空间类别严格大于具有单调开局部有限算子的空间类别。我们证明单调元紧致 GO 空间具有单调开局部有限算子，具有单调（开或不开）闭包保留算子的 GO 空间也是如此，其底层 LOTS 具有 $\sigma$-闭离散密集子集。具有 $\sigma$-闭合离散密集子集和单调闭包保留算子的 GO 空间是可度量化的。具有单调开放闭包保留算子的紧凑 LOTS 是可度量的。