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In this paper, we introduce a new type of closed sets in bitopological space (X, τ1, τ2), used it to construct new types of normality, and introduce new forms of. Definitions. Recall that a topological space is a set equipped with a topological structure. Well, a bitopological space is simply a set equipped. Citation. Patty, C. W. Bitopological spaces. Duke Math. J. 34 (), no. 3, doi/S

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Mathematics Stack Exchange works best with JavaScript enabled. The union of any family of – semiconnected sets with a nonempty intersection is – semiconnected.

The book Bitopological Spaces: Further, we want to find how uniform continuity will work in bitopological spaces. Chandrasekhara Rao and V. Subscribe to Table of Contents Alerts. One of the situations where bitopological spaces occur naturally are asymmetric metric spaces or quasi-metric spaces.

Partial Differential Equations 22 1 73— You could not be signed in. Then, 1 is said to be – open set, if, forthere exists -regular open set such that.

Inthe notion -open sets in bitopological spaces was introduced by Banerjee [ 3 ]. Hence, is – semiconnected. Since is – semicompact, has a finite subcover. A related question was posted at MO: Then and and contains at least one member of and one member of.

Bitopological space – Wikipedia

Such spaces naturally bear two topologies: Quadratic Weyl sums, automorphic functions and invariance principles. Since every – closed set is – semiclosed, is – semiclosed. In my opinion, if some facts which hold in these setting could be stated and proven in a unifying way using bitopological spaces such that spacess of the topologies is finer than the other one, I would personally prefer such formulation even in a paper which deals with only one of these settings.


Then, is called – semiopen set, if there exists an bitooological open set such that -cl.

Concluding Remarks In this paper, some results of – semiconnectedness sspaces compactness in bitopological spaces have been discussed. View at Google Scholar T. Kannan, Department of Mathematics and Statistics, University of Jaffna, Sri Lanka, for providing important references from the literature.

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A situation where two topologies on the same set occur naturally are epireflections and coreflections in the category of topological spaces. If the results fail to hold for bitopology, we are going to illustrate by examples.

Let be a pairwise open cover of. Related articles in Google Scholar. If contains a nonempty proper subset which is both – semiopen and – semiclosed. Definition 6 see [ 6 ]. Now there exist such that also since and also and sincewhich shows that is – semidisconnected subset and it is a contradiction. Then the triple is called a bitopological space.

Thus, is – semiconnected.

An Asymmetric Arzela-Ascoli theoremTopology and nitopological Applications VolumeIssue 11, 1 JunePages — In the realms of applied mathematics and materials science we find many recent applications of asymmetric metric spaces; for example, in rate-independent models for plasticity [6], shape-memory alloys [8], and biyopological for material failure [12].

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Sincewe have. So and are the only subsets of which are both – semiclopen sets. Moreover, we have proved some results in compactness also.

Home Questions Tags Users Unanswered. A cover of a bitopological space is called – open cover ofif andand. Then is called -open, if.

A nonempty collection is called a – semiopen cover of a bitopological spaceif and – – and contains at least one member of – and one member of.

So we are interested to check whether those results will work for bitopological space or not. However, I do not know whether they were studied as bitopologies or whether there are some applications of bitopological spaces in this area.

Let be family of – semiconnected subsets of a bitopological space such that ; then is also – semiconnected. Abstract We are going to establish some results of – semiconnectedness and compactness in a bitopological space. Suppose can be so expressed, then is called – semidisconnected space and we write and it is called – semiseparation of. Arunmaran thanks his supervisor K. Definition 3 see [ 9 ]. Thenwhere is – semiopen, is – semiopen, and are disjoint. But this is impossible. I am interested where bitopological spaces occur in various parts of mathematics i.