Malaya Journal of Matematik, Vol. 6, No. 4, 854-857 2018 https://doi.org/10.26637/MJM0604/0023 On maximal and minimal m-clopen sets in GT spaces Rebati Mohan Roy1* Abstract In this paper, we introduce the notions of maximal and minimal m-clopen sets in a generalized topological space and their some properties. We obtain that maximal and minimal m-clopen sets are independent of maximal and minimal m-open and m-closed sets. We observed that the existence of maximal m-clopen set in a generalized topological space not only ensure the m-disconnectedness of a generalized topological space but also the existence of minimal m-clopen set in that space. Keywords m-open set, m-closed set, maximal m-open set, minimal m-closed set, m-clopen, maximal m-clopen set, minimal m-clopen set. AMS Subject Classification 54A05, 54D05. 1Department of Mathematics, Mathabhanga College, Cooch Behar-736146, West Bengal, India. *Corresponding author: 1roy [email protected] Article History: Received 22 September 2018; Accepted 14 December 2018 c 2018 MJM. Contents denoted by im (A). The generalized closure of a subset A of X is the intersection of all m-closed sets containing A and is 1 Introduction.......................................854 denoted by cm (A). It is easy to see that im (A) = X −cm (X −A). 2 Maximal, minimal m-open set(resp. m-closed set) and By a proper m-open set (resp. m-closed set) of X, we mean m-clopen set ......................................854 a m-open set G (resp. m-closed set E) such that G 6= /0 and 3 Maximal and minimal m-clopen sets . 855 G 6= X if X is m-open (resp. E 6= X and E 6= /0 if /0 is m-closed). We also write R to denote the set of real numbers. References........................................856 We observed that a subset has a double natures in X such as a subset which is both maximal and minimal m-open, then 1. Introduction either X is m-disconnected or number of subset in X is unique There are several generalizations of open sets of topologi- to certain condition, e. g. theorem 2.4 [8]. Due to theorem cal spaces. Two well-discussed generalizations of open sets 3.10 of this paper we obtain that the existence of a set which of topological spaces are semi-open [4] and pre-open sets [5]). both maximal and minimal m-clopen not only assures the m- It is observed that the semi-open, pre-open sets of a topolog- disconnectedness of space but also ensure the existence of ical space possess properties resembling those of open sets another set which is also both maximal m-clopen and minimal of the topological space with some exception e.g. the family m-clopen and these two are only m-clopen sets in that space. of semi-open (or pre-open) sets are not closed even under This notion reveals the difference among of maximal and min- finite intersections. Starting from this observation, Csasz´ ar´ imal m-open sets which the maximal and minimal m-clopen [1] finally introduced and studied the concepts of generalized and also beauty of maximal and minimal m-clopen sets. topology. Let X be a nonempty set and m be a subcollection of the 2. Maximal, minimal m-open set(resp. power set of X. Then m is a generalized topology on X if /0 2 m and m is closed under unions. We write, GT (resp. GT m-closed set) and m-clopen set space) to denote a generalized topology (resp. generalized We recall the following known definitions and results to topological space) for brevity. The members of m are called make the article self sufficient as far as practical. the m-open sets in X and the complement of a m-open set is called a m-closed set in X. The generalized interior of a subset Definition 2.1 (Roy and Sen [6]). A proper m-open set A of A of X is the union of all m-open sets contained in A and is a GT space X is called a maximal m-open set if there is no On maximal and minimal m-clopen sets in GT spaces — 855/857 m-open set U(6= A;X) such that A ⊂ U ⊂ X. 3. Maximal and minimal m-clopen sets Definition 2.2 (Roy and Sen [6]). A proper m-closed set E of We now introduce some new notions and obtain their follow- a GT space X is called a minimal m-closed set if there is no ing properties . m-closed set F(6= /0;E) such that /0 ⊂ F ⊂ E. Definition 3.1. A proper m-clopen set U of a GT space X is Definition 2.3 (S. Al Ghour et al. [7]). A proper m-open set said to be minimal m-clopen if V is a proper m-clopen set U of X is said to be a minimal m-open set if the only proper such that V ⊂ U, then V = U or V = /0. m-open set which is contained in U is U. Example 3.2. Let X = (−1;1) and m = f/0;X;f0g;(−1;0); Definition 2.4 (Mukharjee [8]). A proper m-closed set E in a (−1;0], (0;1);[0;1);X − f0gg. GT space X is called a maximal m-closed set if any m-closed set which contains E is X or E. In the GT space (X; m), [0;1) is a minimal m-clopen set but it is neither minimal m-open nor minimal m-closed. Definition 2.5 (Mukharjee [8]). A proper subset of a GT We observed that a minimal m-open or a minimal m-closed space X is said to be m-clopen if it is both m-open and m- set may not be minimal m-clopen. So the notion of minimal closed. m-clopen set is independent to the notion of minimal m-open Definition 2.6 (Csasz´ ar´ [3]). A GTspace X is m-connected if as well as minimal m-closed sets. It is clearly to see that if X can not be expressed as G[H = X where G;H are disjoined a set A is both minimal m-open and minimal m-closed, then m-open sets. As usual, if X is not m-connected then X is called A is minimal m-clopen. In fact, a m-clopen set is minimal m-disconnected. So X is m-disconnected if there exists two m-clopen if it is either minimal m-open or minimal m-closed. disjoint m-open sets G;H such that G [ H = X. Definition 3.3. A proper m-clopen set U of a GT space X is Theorem 2.7 (Roy and Sen [6]). If A is a maximal m-open set said to be maximal m-clopen if V is a proper m-clopen set and B is a m-open set in a GT space X, then either A [ B = X such that V ⊂ U, then V = U or B = X. or B ⊂ A. If B is also a maximal m-open set distinct from A, (X; ) (− ; ) then A [ B = X. In the GT space m , Example 3.2, 1 0 is a maximal m-clopen set but it is neither maximal m-open nor maximal Theorem 2.8 (Roy and Sen [6]). If F is a minimal m-closed m-closed. set and E is a m-closed set in a GT space X, then either We observed that a maximal m-open or a maximal m-closed E \ F = /0 or F ⊂ E. If E is also a minimal m-closed set set may not be maximal m-clopen. So, the notion of maximal distinct from F, then F \ E = /0. m-clopen set is independent to the notion of maximal m-open as well as maximal m-closed sets. It is clearly to see that if Theorem 2.9 (Mukharjee [8]). If U is a minimal m-open set a set A is both maximal m-open and maximal m-closed, then and W is a m-open set such that U \W is a m-open set, then A is maximal m-clopen. In fact, a m-clopen set is maximal either U \W = /0 or U ⊂ W. If W is also a minimal m-open m-clopen if it is either maximal m-open or maximal m-closed. set distinct from U, then U \W = /0. In [8] we analyzed that if a GT space has only one proper -open set, then it is both maximal and minimal -open. But Theorem 2.10 (Mukharjee [8]). If G is a maximal m-open m m we note that it is neither maximal -clopen nor minimal - set and H is a minimal m-open set in a GT space X such m m clopen. If a GT space X has only two proper -open sets that G \ H is a m-open set then either H ⊂ G or the space is m such that one is not contain in other, then both are maximal m-disconnected. m-clopen and minimal m-clopen. Also maximal m-clopen Theorem 2.11 (Mukharjee [8]). If E is a maximal m-closed or minimal m-clopen sets can exists in a m-disconnected GT set and F is any m-closed set in a GT space X such that E [F space.