We consider an alternate form of the equivalence between the category of compact Hausdorff spaces and continuous functions and a category formed from Gleason spaces and certain relations. This equivalence arises from the study of the projective cover of a compact Hausdorff space. This line leads us to consider the category of compact Hausdorff spaces with closed relations, and the corresponding subcategories with continuous and interior relations. Various equivalences of these categories are given extending known equivalences of the category of compact Hausdorff spaces and continuous functions with compact regular frames, de Vries algebras, and also with a category of Gleason spaces that we introduce. Study of categories of compact Hausdorff spaces with relations is of interest as a general setting to consider Gleason spaces, for connections to modal logic, as well as for the intrinsic interest in these categories.

中文翻译：

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具有关系和格里森空间的紧致豪斯多夫空间

我们考虑紧致豪斯多夫空间和连续函数的范畴与由格里森空间和某些关系形成的范畴之间的等价的另一种形式。这种等价源于对紧致豪斯多夫空间的射影覆盖的研究。这条线引导我们考虑具有封闭关系的紧致豪斯多夫空间的范畴，以及具有连续和内部关系的相应子范畴。给出了这些类别的各种等价，扩展了紧凑 Hausdorff 空间类别和具有紧凑规则框架的连续函数、de Vries 代数以及我们介绍的格里森空间类别的已知等价。研究具有关系的紧致 Hausdorff 空间的类别作为考虑 Gleason 空间的一般设置很有趣，