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Compact Hausdorff Spaces with Relations and Gleason Spaces

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Abstract

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.

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Acknowledgements

We thank the referee for pointing out [19] to us, as well as for a number of useful suggestions, particularly involving adjunctions in order enriched categories.

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Authors and Affiliations

  1. New Mexico State University, Las Cruces, USA

    G. Bezhanishvili & J. Harding

  2. TSU Razmadze Mathematical Institute, Tbilisi, Georgia

    D. Gabelaia & M. Jibladze

Authors
  1. G. Bezhanishvili
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  2. D. Gabelaia
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  3. J. Harding
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  4. M. Jibladze
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Correspondence to G. Bezhanishvili.

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Communicated by Jorge Picado.

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Bezhanishvili, G., Gabelaia, D., Harding, J. et al. Compact Hausdorff Spaces with Relations and Gleason Spaces. Appl Categor Struct 27, 663–686 (2019). https://doi.org/10.1007/s10485-019-09573-x

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  • DOI: https://doi.org/10.1007/s10485-019-09573-x

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