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2018, Afrika Matematika
We see that the real numbers system with the usual topology contains no minimal open sets. This observation instigates us to study topological spaces having no minimal and maximal open sets. We find that such topological spaces if connected are not cut-point spaces. We also characterize mean open sets in T 1 connected topological spaces.
In this paper a new class of topological spaces called T min spaces and T max spaces and study their relations with topological spaces. Also a new class of maps called minimal continuous, maximal continuous, minimal irresolute, maximal irresolute, minimalmaximal continuous and maximal-minimal continuous maps in topological spaces and study their relations with various types of continuous maps. 2000 MATHEMATICS CLASSIFICATION: 54C05 Key words and phrases: Minimal open sets and Maximal open sets.
In this paper, we introduce and define minimal-open sets in topological spaces and we obtain some basic properties of this set. Moreover, we define-locally finite space and give some applications for finite minimal-open sets.
Mathematics and Statistics, 2017
The purpose of this paper is to investigate the concepts of minimal and maximal regular open sets and their relations with minimal and maximal open sets. We study several properties of such concepts in a semi-regular space. It is mainly shown that if X is a semi-regular space, then m i O(X) = m i RO(X). We introduce and study new type of sets called minimal regular generalized closed. A special interest type of topological space called rT min space is studied and obtain some of its basic properties.
International Journal of Mathematics Trends and Technology, 2017
In 2008, Caldas M, Jafari S. and Noiri T. [7] introduced the concept of maximal -open sets, minimal -closed sets, -semi-maximal open and semi-minimal closed sets in general topological settings. In the present paper a new class of sets called minimal -open sets and maximal -closed sets in a topological space are introduced which are the -open sets and -closed sets respectively. The complement of minimal -open set is a maximal closed set. Some properties of -semi maximal closed sets, -semi minimal open sets are studied. Keywords-Minimal -open set, Maximal -closed set, -semi-minimal open set, -semi-maximal closed set.
The American Mathematical Monthly, 1969
have defined a point x in a topological space (X, 9~) to be a strong cut point if the set X -{x} has two components. They then show that " a connected and locally connected separable and regular space, in which every point is a strong cut point, is homeomorphic to the real line " [1]. We will show that a separable Hausdorjf space (X, ST) is homeomorphic to the real numbers if every xeX is a strong cut point and the set of components of complements of point sets forms a subbasefor the space (X, ^). We shall call a topological space satisfying the conditions of this theorem a cut space or C-space. The subbase we denote by £f and if A £ X then we denote the boundary of A by B(A). We now proceed with a sequence of lemmas.
2012
In this paper, the notion of maximal m-open set is introduced and its properties are investigated. Some results about existence of maximal m-open sets are given. Moreover, the relations between maximal m-open sets in an m-space and maximal open sets in the corresponding generated topology are considered. Our results are supported by examples and counterexamples.
In this paper, a new class of open sets called R # -open sets in topological spaces are introduced and studied. This new class of open sets lies between the gopen sets and rg-open sets in topological spaces. Also some of their properties have been investigated.
The notion of maximal and minimal open sets in a topological space was introduced by [4] and [5]. In this paper, we introduce new classes of sets called maximal semi-open sets and minimal semi-open sets and investigate some of their fundamental properties. 2000 Mathematics Subject Classification: Primary: 54A05, 54A10; Secondary: 54E55. A.B.Khalaf and H.M.Hasan -On Some New Maximal and Minimal ...
IAEME Publication, 2020
In this paper a new class of minimal open and minimal closed sets in topological spaces, namely minimal-open and minimal-closed sets are introduced. We give some basic properties and various characterizations of minimal-open and minimalclosed sets.
International Journal of Mathematics Trends and Technology, 2017
E. Ekici [8] introduced e-open (resp. eclosed) sets in general topology. Thereafter Nakaoka and Oda ([1] and [2]) initiated the notion of maximal open (resp. minimal closed) sets in topological spaces. In the present work, the author introduces new classes of open and closed sets called maximal e-open sets, minimal e-closed sets, esemi maximal open and e-semi minimal closed and investigate some of their fundamental properties with example and counter examples.
Journal of Al-Nahrain University Science, 2011
In this work we introduce maximal m-open set in minimal structure spaces and study some of their basic properties in these spaces.
Kyungpook mathematical journal, 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.
Hacettepe Journal of Mathematics and Statistics, 2017
In this paper, we obtain some more properties of mean open and closed sets. We also introduce a notion of mean clopen sets and obtain some properties of such sets. We observe that a mean clopen set is both mean open and mean closed but the converse is not true.
International Mathematical Forum
In this paper, we introduce a new class of open sets in a topological space called − open sets. We study some properties and several characterizations of this class, also we explain the relation of − open sets with many other classes of open sets. Furthermore, we define − closed sets and − closed sets and we give some fundamental properties and relations between these classes and other classes such as − closed and − closed sets.
International journal of scientific research in computer science, engineering and information technology, 2017
In this work, some conditions for -disconnectedness of a topological space in terms of maximal and minimal δθopen sets and also some similar results in terms of maximal and minimal δθ-closed sets and also interrelationships between them are investigated. Generally, we find that if a space has a set which is both maximal and minimal δθopen, then either this set is the only non-trivial δθ-open set in the space or the space is δθ-disconnected. We also obtain a result concerning a minimal δθ-open set on a subspace.
We introduce the notion of minimal open sets in a generalized topological space (X, µ). We investigate some of their fundamental properties and proved that any subset of a minimal open set on a GTS (X, µ) is a µ-preopen set.
Malaya Journal of Matematik, 2020
In this paper, we introduce and study cleanly µ-covered spaces along with two strong separation axioms in generalized topological spaces. Strong separation axioms are investigated by means of minimal µ-open and µ-closed sets of generalized topological spaces. Keywords µ-open set, µ-closed set, maximal µ-open set, minimal µ-open set, cleanly µ-covered.
The American Mathematical Monthly, 1969
Acta Mathematica Hungarica, 2012
We introduce the notion of maximal μ-open and minimal μclosed sets in a generalized topological space. We also investigate some of their fundamental properties.
2008
We construct an example of a real-valued continuous non-constant function $f$ defined on a connected complete metric space $X$ such that every point of $X$ is a point of local minimum or local maximum for $f$. The space $X$ is connected but fails to be separably connected.
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