The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $$\textsf {Top}$$ Top of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class $$\mathcal {C}$$ C of objects we generalize the notion of $$\mathcal {C}$$ C -generated spaces , from which we derive, for instance, a general concept of Alexandroff spaces . Furthermore, as done for $$\textsf {Top}$$ Top , we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

中文翻译：

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拓扑中的紧致空间和拟空间

广义丰富类别的紧致性和 Hausdorff 分离的概念使我们能够像经典的类别 $$\textsf {Top}$$ Top of Topological Spaces 和连续函数一样，在这种情况下研究紧凑生成的空间和准空间。此外，对于类 $$\mathcal {C}$$ C 的对象，我们概括了 $$\mathcal {C}$$ C 生成空间的概念，例如，我们从中导出了 Alexandroff 空间的一般概念. 此外，正如对 $$\textsf {Top}$$ Top 所做的那样，我们还在我们的一般性水平上研究了紧凑生成空间和准空间之间的关系。