They are described by cover form and finite intersection property. Journal of mathematical analysis and applications 162, 92110 1991 a generalised form of compactness in fuzzy topological spaces j. On rgbsets in a generalized topological space 1795 theorem 2. A generalised form of compactness in fuzzy topological spaces.
Here is another useful property of compact metric spaces, which will eventually be generalized even further, in e below. A base for an lfuzzy topology f on a set x is a collection 93 c y. Compactness in fuzzy topological spaces 549 and u v v and a9 are similarly defined. In this paper, a new notion of sequential compactness is introduced in ltopological spaces, which is called sequentially s. Somewhat compactness and somewhat connectedness in topological spaces dr. A generalized topology in a set x is a collection cov x of families of subsets of x such that the triple x. Preliminaries throughout this paper x,w, y,v are topological spaces with no separation axioms. If the subset f of cx,y is totally bounded under the uniform metric corresponding to d, then f is equicontinuous under d.
However, some authors investigated generalizations of the notion of topology. Theconceptsof compactness, countable compactness, the lindel. The aim of this paper is to introduce a new notion of lfuzzy compactness in lfuzzy topological spaces, which is a generalization of lowens fuzzy compactness in l topological spaces. Some types of semicompactness via hereditary classes 1. In general, though, does the metric definition imply the topological definition. The aim of this paper is to continue the study of sgcompact spaces, a topologi. If l0,1, sequential ultracompactness, sequential ncompactness and sequential strong compactness imply sequential s. In this paper, a new notion of compactness is introduced in ltopological spaces by means of betaaopen cover and qaopen cover, which is called s compactness. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. As a sort of converse to the above statements, the preimage of a compact space under a proper map is compact. Introduction in chapter i we looked at properties of sets, and in chapter ii we added some additional structure to a set a distance function to create a pseudomet. Why is this difference regarding the terminology of compactness.
In mathematics, more specifically in general topology, compactness is a property that. In this work, notions of topological and admissible compactness of general ized topologies are introduced to begin and investigate a theory of compacti. In 1965 26, zadeh generalized the concept of a crisp set by introducing the concept of a fuzzy set. A subset s of a topological space x is relative compact when the closure clx is compact. Furthermore, we study its several properties and characterizations in detail. Metric spaces have many nice properties, like being rst countable, very separative, and so on, but compact spaces facilitate easy proofs. View on compactness and connectedness via semi generalized b. Grwo generalized regular weakly open compactness, and grwconnectedness in a topological spaces a. The purpose of the present paper is to introduce the concepts of.
On g r connectedness and g r compactness in topological. In this work, a new notion of a strict compactification of a generalized topological space is introduced and investigated in zf. It is well known that the concept of compactness and connectedness plays an important role in generalized topological space. We then looked at some of the most basic definitions and properties of pseudometric spaces. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. The complement of sgclosed set is said to be sgopen set. The aim of this paper is to introduce the concept of g rconnected and g r compactness in topological spaces. In this paper, we obtain a new generalization of closed sets in the weaker topological space x. Baiju and sunil jacob john 17introduced a class of submetacompactness in ltopological spaces. Chapter 6 compactness and connectedness of semi generalized bopen sets in topological spaces 6. The aim of this paper is to introduce and study strong forms of u compactness in generalized topological spaces with respect to a hereditary class, called suh compactness and s suh compactness. On fuzzy compactness in intuitionistic fuzzy topological spaces. This is not compact, but if fu igis an open cover then we can \re ne it in.
Characterizations of compact metric spaces france dacar, jo. In this paper, rcontinuous functions and r compactness are introduced in ideal topological spaces and analyzed the relationships with continuous functions and compactness in general topological spaces. Similarly, forany opencovering of, if there is a finite subcovering, then is called a compact set and also the compact upper. We also study some properties of lfuzzy f compactness.
On fuzzy continuity in intuitionistic fuzzy topological spaces. Compactness 1 motivation while metrizability is the analysts favourite topological property, compactness is surely the topologists favourite topological property. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. After the discovery of fuzzy sets, much attention has. Compactness in lsmooth ideal topological spaces 1647. In this work, notions of topological and admissible compactness of generalized topologies are introduced to begin and investigate a theory of compacti. Notes on compactness version of th september 2005 martn escard o. A comparison between these types and some di erent forms of compactness in fuzzy topology is established. A subset aof a topological space x, is called sgclosed, if scla u. Somewhat compactness and somewhat connectedness in. Jafari et al 18 studied gocompact spaces and kupka type func. Sep 09, 2019 on fuzzy compactness in intuitionistic fuzzy topological spaces. The family of all sgbopen respectively sgbclosed sets of x,t is denoted by sgb ox,t.
Pdf theory of generalized compactness in generalized. Ont generalized pre connectedness andt generalized pre. Slightly more generally, this is true for an upper semicontinuous function. Compact fuzzy topological spaces were first introduced in the literature by chang l who proved two results about such spaces. In 1988, fuzzy compactness, strong compactness and ultracompactness were generalized to general lfuzzy subset by wang in 17 these can also be seen in 9. Rperfect sets, ropen sets, rcontinuous functions, r compactness 1 introduction and preliminaries. Mugarjee and roy 2007 discussed a new type of compactness via grills.
To prove the converse, it will su ce to show that e b. The second chapter deals with properties of compact spaces, conditions equivalent to compactness, and the relationship of these ideas with the separation properties. To get started, consider rn with its usual topology. Rperfect sets, ropen sets, rcontinuous functions, r compactness 1 introduction and preliminaries a non empty collection of subsets of a set x is said to be an ideal on x, if it. The aim of this paper is to introduce a new notion of lfuzzy compactness in lfuzzy topological spaces, which is a generalization of lowens fuzzy compactness in ltopological spaces. A base for an lfuzzy topology f on a set x is a collection 93 c y such that, for each u e r there exists gu c g. Devi et al 12 introduced a class of generalized semi opencompact and semi generalized opencompact in topological spaces and investigated some of its theorems. Metricandtopologicalspaces university of cambridge. For any open covering of, if there is a finite subcovering, then is called a compact set and also the compact lower approximation of rx. Note that relative compactness does not carry over to topological subspaces. Metric spaces, topological spaces, and compactness 255 theorem a. The particular cases of sequential compactness and of. Connectedness and compactness is one of the most important and fundamental concepts in topology.
The idea of regarding functions as themselves points of a generalized space dates back to the. The concept of compactness of an itopological space was first. The aim of this paper is to introduce the concept of g rconnected and g rcompactness in topological spaces. Compactness in metric spacescompact sets in banach spaces and hilbert spaceshistory and motivationweak convergencefrom local to globaldirect methods in calculus of variationssequential compactnessapplications in metric spaces equivalence of compactness theorem in metric space, a subset kis compact if and only if it is sequentially compact. Cov x, cov x is a generalized topological space a gts in the sense of delfs and knebusch. We also investigate the relationships between fuzzy semi alpha almost compactness and fuzzy semi alpha nearly compactness.
Fuzzy lterbases are used to characterize these concepts. Moreover, we can call a strict subspace of a gts admissibly compact if this. Metric spaces, topological spaces, and compactness 253 given s. Throughout this paper x and y are topological spaces on which no separation axioms are assumed unless otherwise explicitly stated. Motivation there is an extraordinarily useful weaking of compactness that is satis ed by virtually all ice topological spaces that arise in geometry and analysis. Chapter 6 compactness and connectedness of semi generalized b. Jafari and rajesh 17 investigated the weaker and stronger forms of girresolute functions. In this paper we introduce anew form of soft supra compact spaces namely, soft supra compact spaces, soft supra closed spaces, soft supra lindel of spaces and soft supra generalized compactness. In this paper, we give the concept of lfuzzy fopen operator in lfuzzy topological spaces, and use it to introduce a new form of f compactness in lfuzzy topological spaces. The family of all sgbopen respectively sgbclosed sets of x,t is denoted by sgb ox,t respectively sgb clx,t. Chadwick department of mathematics pure and applied, rhodes university, grahamstown, south africa submitted by ulrich hle received september 27, 1990 the a compactness of wang, extended in 15, provides the first theory of compactness which is. Topologycompactness wikibooks, open books for an open world. Nachbin 6 observed that, more generally, the intersection of compactly many open sets is open see section 2 for a precise formulation of this fact.
A new form of fcompactness in lfuzzy topological spaces. P for generalizing 15456 types of compactness and closeness. Compactness and compactifications in generalized topology. Chadwick department of mathematics pure and applied, rhodes university, grahamstown, south africa submitted by ulrich hle received september 27, 1990 the a compactness of wang, extended in 15, provides the first theory of compactness. A generalised form of compactness in fuzzy topological spaces j. We prove that the fuzzy unit interval is txcompact. Uft holds if and only if all wallman extensions of every weakly normal generalized topological space are compact. The purpose of this study is to define and study the notion of. In 2016 baker introduced the notion of somewhat open set in topological space and used it. Chadwick department of mathematics pure and applied, rhodes university, grahamstown, south. As a result, it is possible to have fuzzy sets which are never ncompact, even if the fuzzy topology has only a finite number of open fuzzy sets. In this paper, a new notion of compactness is introduced in l topological spaces by means of betaaopen cover and qaopen cover, which is called s compactness. Some branches of mathematics such as algebraic geometry, typically influenced by the french school of bourbaki, use the term quasicompact for the general notion, and reserve the term compact for topological spaces that are both hausdorff and quasicompact.
While compact may infer small size, this is not true in general. We need one more lemma before proving the classical version of ascolis theorem. Compactness in pythagorean fuzzy topological spaces asian. Countable compactness in generalized ltopological spaces. The aim of this paper is to introduce the concept of generalized pre connectedness and generalized pre compactness in topological spaces. In this paper, we use the concept of generalized topological spaces introduced by csaszar in. Generalized topological semantics for weak modal logics. Let x be a topological space and let y,d be a metric space. In this paper, the generalized countable lcompact sets and generalized lindelof sets are introduced in generalized ltopological spaces, based on the notion of generalized lcompactness. We obtain a tychonoff theorem for an arbitrary product of orcompact fuzzy spaces and a lpoint compactification. A generalised form of compactness in fuzzy topological. A closed subset of a compact space is compact, and a finite union of compact sets is compact.
For this reason, it is difficult to speak about topological semantics for logics weaker than s4, not to mention nonnormal systems. Throughout this paper we shall be concerned with real linear topological spaces x, although our results can be obtained for. Namely, we will discuss metric spaces, open sets, and closed sets. Pushpalatha department of mathematics, government arts and science college, udumalpet, india p. For a generalized topological space x, the elements of are called open sets, the complements of open sets are called closed sets, the union of all elements of will be denoted. On semi generalized star b connectedness and semi generalized star b compactness in topological spaces called sgbopen. Since the concept of fuzzy sets corresponds to the phys. Box 1664 al khobar 31952, kingdom of saudi arabia abstract. Metric spaces, topological spaces, and compactness. Fuzzy topological spaces, fuzzy compactness, fuzzy closed spaces. For example, the open interval 0,1 is relatively compact in r with the usual topology, but is not relatively compact in itself. Topological definition of compactness implies metric. Pdf we define and study the notion of compact space on generalized topological spaces. Theory of generalized sets in generalized topological spaces.
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