Korean J. Math. Vol. 29 No. 3 (2021) pp.493-499
DOI: https://doi.org/10.11568/kjm.2021.29.3.493

On maximal compact frames

Main Article Content

Jayaprasad P N
Madhavan Namboothiri N M
Santhosh P K
Varghese Jacob

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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References

[1] A.Ramanathan, Minimal-bicompact spaces, J.Indian Math.Soc.12(1948), 40–46. Google Scholar

[2] B.Banaschewski, Singly generated frame extensions, J.of Pure.Appl.Alg., 83(1992), 1–21. Google Scholar

[3] C.H.Dowker and D.Strauss, Sums in the category of frames, Houston J.Math., 3(1977), 7–15. Google Scholar

[4] Jayaprasad, P. N. and T.P.Johnson, Reversible Frames, Journal of Advanced Studies in Topology, Vol.3, No.2(2012), 7–13. Google Scholar

[5] Jayaprasad. P.N, On Singly Generated Extension of a Frame, Bulletin of the Allahabad Math. Google Scholar

[6] Soc., No.2, 28(2013), 183–193. Google Scholar

[7] J.Paseka. and B.S ̆marda, T2 Frames and Almost compact frames, Czech.Math.J.42(1992), 385– Google Scholar

[8] Google Scholar

[9] J.Picado and A.Pultr, Frames and Locales-Topology without Points, Birkha ̈user, 2012. Google Scholar

[10] J.R. Isbell, Atomless parts of spaces, Math.Scand., 31(1972), 5–32. Google Scholar

[11] N.Levine, When are compact and closed equivalent?, Amer.Math.Month., No,1, 71(1965), 41–44. Google Scholar

[12] P.T.Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics 3, Camb. Univ.Press, 1982. Google Scholar