• Title/Summary/Keyword: maximal open set

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ON MAXIMAL, MINIMAL OPEN AND CLOSED SETS

  • MUKHARJEE, AJOY
    • Communications of the Korean Mathematical Society
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    • v.30 no.3
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    • pp.277-282
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    • 2015
  • We obtain some conditions for disconnectedness of a topological space in terms of maximal and minimal open sets, and some similar results in terms of maximal and minimal closed sets along with interrelations between them. In particular, we show that if a space has a set which is both maximal and minimal open, then either this set is the only nontrivial open set in the space or the space is disconnected. We also obtain a result concerning a minimal open set on a subspace.

MORE ON MAXIMAL, MINIMAL OPEN AND CLOSED SETS

  • Mukharjee, Ajoy
    • Communications of the Korean Mathematical Society
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    • v.32 no.1
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    • pp.175-181
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    • 2017
  • In this paper, we introduce a notion of cleanly covered topological spaces along with two strong separation axioms. Some properties of cleanly covered topological spaces are obtained in term of maximal open sets including some similar properties of a topological space in term of maximal closed sets. Two strong separation axioms are also investigated in terms of minimal open and maximal closed sets.

MORE ON FUZZY MAXIMAL, MINIMAL OPEN AND CLOSED SETS

  • SWAMINATHAN, A.;SIVARAJA, S.
    • Journal of applied mathematics & informatics
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    • v.39 no.3_4
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    • pp.251-257
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    • 2021
  • This article is devoted to introduce the notion of fuzzy cleanly covered fuzzy topological spaces; in addition two strong fuzzy separation axioms are studied. By means of fuzzy maximal open sets some properties of fuzzy cleanly covered fuzzy topological spaces are obtained and also by means of fuzzy maximal closed sets few identical results of a fuzzy topological spaces are investigated. Through fuzzy minimal open and fuzzy maximal closed sets, two strong fuzzy separation axioms are discussed.

ON FUZZY MAXIMAL, MINIMAL AND MEAN OPEN SETS

  • SWAMINATHAN, A.;SIVARAJA, S.
    • Journal of Applied and Pure Mathematics
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    • v.4 no.1_2
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    • pp.79-84
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    • 2022
  • We have observed that there exist certain fuzzy topological spaces with no fuzzy minimal open sets. This observation motivates us to investigate fuzzy topological spaces with neither fuzzy minimal open sets nor fuzzy maximal open sets. We have observed if such fuzzy topological spaces exist and if it is connected are not fuzzy cut-point spaces. We also study and characterize certain properties of fuzzy mean open sets in fuzzy T1-connected fuzzy topological spaces.

On Mean Open and Closed Sets

  • Mukharjee, Ajoy;Bagchi, Kallol Bhandhu
    • Kyungpook Mathematical Journal
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    • v.56 no.4
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    • pp.1259-1265
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    • 2016
  • In this paper, we introduce the notions of mean open and closed sets in topological spaces, and obtain some properties of such sets. We observe that proper paraopen and paraclosed sets are identical to mean open and closed sets respectively.

SOME REMARKS ON FUZZY MEAN OPEN, CLOSED AND CLOPEN SETS

  • SWAMINATHAN, A.;SANKARI, M.
    • Journal of applied mathematics & informatics
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    • v.39 no.5_6
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    • pp.743-749
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    • 2021
  • The purpose of this article is to study few properties of fuzzy mean open and fuzzy mean closed sets in fuzzy topological spaces. Further, the idea of fuzzy mean clopen set is introduced. It is observed that a fuzzy mean clopen set is both fuzzy mean open and fuzzy mean closed but the converse is not true.

OPENLY SEMIPRIMITIVE PROJECTIVE MODULE

  • Bae, Soon-Sook
    • Communications of the Korean Mathematical Society
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    • v.19 no.4
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    • pp.619-637
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    • 2004
  • In this paper, a left module over an associative ring with identity is defined to be openly semiprimitive (strongly semiprimitive, respectively) by the zero intersection of all maximal open fully invariant submodules (all maximal open submodules which are fully invariant, respectively) of it. For any projective module, the openly semiprimitivity of the projective module is an equivalent condition of the semiprimitivity of endomorphism ring of the projective module and the strongly semiprimitivity of the projective module is an equivalent condition of the endomorphism ring of the projective module being a sub direct product of a set of subdivisions of division rings.

INTERIORS AND CLOSURES IN A SET WITH AN OPERATION

  • Nakaoka, Fumie;Oda, Nobuyuki
    • Communications of the Korean Mathematical Society
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    • v.29 no.4
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    • pp.555-568
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    • 2014
  • A set with an operation defined on a family of subsets is studied. The operation is used to generalize the topological space itself. The operation defines the operation-open subsets in the set. Relations are studied among two types of the interiors and the closures of subsets. Some properties of maximal operation-open sets are obtained. Semi-open sets and pre-open sets are defined in the sets with operations and some relations among them are proved.