Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON MAXIMAL COMPACT FRAMES

  • Received : 2021.02.12
  • Accepted : 2021.07.06
  • Published : 2021.09.30
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Abstract

Every closed subset of a compact topological space is compact. Also every compact subset of a Hausdorff topological space is closed. It follows that compact subsets are precisely the closed subsets in a compact Hausdorff space. It is also proved that a topological space is maximal compact if and only if its compact subsets are precisely the closed subsets. A locale is a categorical extension of topological spaces and a frame is an object in its opposite category. We investigate to find whether the closed sublocales are exactly the compact sublocales of a compact Hausdorff frame. We also try to investigate whether the closed sublocales are exactly the compact sublocales of a maximal compact frame.

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Acknowledgement

We are grateful to the valuable suggestions of Dr. T. P Johnson, Professor and Head, Department of Applied Sciences and Humanities, School of Engineering, Cochin University of Science and Technology, Cochin, Kerala, India for improving this paper. We also thank the anonymous reviewers for their valuable suggestions which resulted in the improvement of the paper.

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