Korean Journal of Mathematics

  • 제29권3호
  • /
  • Pages.493-499
  • /
  • 2021
  • /
  • 1976-8605(pISSN)
  • /
  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON MAXIMAL COMPACT FRAMES

  • 투고 : 2021.02.12
  • 심사 : 2021.07.06
  • 발행 : 2021.09.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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.

키워드

과제정보

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.

참고문헌

  1. A.Ramanathan, Minimal-bicompact spaces, J.Indian Math.Soc.12(1948), 40-46.
  2. B.Banaschewski, Singly generated frame extensions, J.of Pure.Appl.Alg., 83(1992), 1-21. https://doi.org/10.1016/0022-4049(92)90101-K
  3. C.H.Dowker and D.Strauss, Sums in the category of frames, Houston J.Math., 3(1977), 7-15.
  4. Jayaprasad, P. N. and T.P.Johnson, Reversible Frames, Journal of Advanced Studies in Topology, Vol.3, No.2(2012), 7-13.
  5. Jayaprasad. P.N, On Singly Generated Extension of a Frame, Bulletin of the Allahabad Math. Soc., No.2, 28(2013), 183-193.
  6. J.Paseka. and B.Smarda, T2 Frames and Almost compact frames, Czech.Math.J.42(1992), 385-402. https://doi.org/10.21136/CMJ.1992.128349
  7. J.Picado and A.Pultr, Frames and Locales-Topology without Points, Birkhauser, 2012.
  8. J.R. Isbell, Atomless parts of spaces, Math.Scand., 31(1972), 5-32. https://doi.org/10.7146/math.scand.a-11409
  9. N.Levine, When are compact and closed equivalent?, Amer.Math.Month.,No,1, 71(1965), 41-44. https://doi.org/10.2307/2312998
  10. P.T.Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics 3, Camb.Univ.Press, 1982.