In thispaper, an attempt is made to study about compactness and connectedness in rough topological spaces. If you think that we should keep this tag, feel free to bring this up in tagging chatroom or on meta. Also some characterisations concerning the igpr connectedness and igpr connected sets were studied and compared. Metric characterisation of connectedness for topological spaces. Metricandtopologicalspaces university of cambridge. Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple connectedness. More connectedness in topological spaces 75 aho, nieminen, popa, noiri, and jafari have studied semipreconnectedness. Stronger and weaker forms of connectedness have been introduced and investigated by noiri 3, reilly et al. Request pdf metric characterization of connectedness for topological spaces connectedness, path connectedness, and uniform connectedness are wellknown concepts. Furthermore, the notion of connectedness degrees of fuzzy subsets is introduced. Connectedness in fuzzy soft topological spaces springerlink.
Based on the consideration to the layer structures of fuzzy lattices and the level topologies of lfuzzy topological spaces, a connectedness is define. On fuzzy modular spaces shen, yonghong and chen, wei, journal of applied mathematics, 20. In contrast to connectedness of topological spaces, if a topology. Also, we introduce and study the notion of connectedness to fuzzy soft topological spaces with fuzzy soft grills. Pdf igpr connectedness on intuitionistic topological spaces. Extension lectures connectedness and disconnectedness in. Recently many fuzzy topological concepts such as fuzzy connectedness have been generalized in. In 2016 baker introduced the notion of somewhat open set in topological space and used it. We will allow shapes to be changed, but without tearing them. Conversely, the only topological properties that imply is connected are very extreme such as 1 or \ l\ly\ has the trivial topology.
Connectedness 1 motivation connectedness is the sort of topological property that students love. Somewhat compactness and somewhat connectedness in. We present a twofold generalization of this result by eliminating compactness and considering flaggs continuity spaces. A topological space x is said to be hyperconnected 11 if every pair of nonempty open sets of x has nonempty intersection.
Somewhat compactness and somewhat connectedness in topological spaces dr. Further closed sets like i rg,i rw were further developed by navaneethakrishnan 10 and a. Caspian journal of mathematical sciences cjms, 2019. Closed sets, hausdor spaces, and closure of a set 9 8. Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces. A study of compactness and connectedness in rough topological. Another example of connectivity can be found in regular tilings. Metric characterization of connectedness for topological spaces. Connectedness in topological spaces can also be defined in terms of chains governed by open coverings in a manner that is more reminiscent of path connectedness. Additionally, connectedness and pathconnectedness are the same for finite topological spaces. Introduction in this chapter we introduce the idea of connectedness. Devi 17 brought the concept complex topological space and investigated some properties of. Connectedness and some concepts related to connectedness of a.
The particular distance function must satisfy the following conditions. Part ii is an introduction to algebraic topology, which associates algebraic structures such as groups to topological spaces. Then we call k k a norm and say that v,k k is a normed vector space. Thus we must insist that all our theorems be generalizations of wellknown results for connectedness in topological spaces. Many properties of connectedness in general topology are generalized to fuzzy topological spaces 1. Roughly speaking, a connected topological space is one that is \in one piece. Neutrosophic complex connectedness in neutrosophic complex. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. Namely, we will discuss metric spaces, open sets, and closed sets.
Xyis a continuous map between topological spaces, xis connected, yis hausdor. Topology underlies all of analysis, and especially certain large spaces such. Notations the following notations are used in this paper. A rigorous introduction to set theory, topological spaces, continuity, separation, metrizability, compactness, connectedness, function spaces, and algebraic topology kindle edition by warner, steve. Topologyconnectedness wikibooks, open books for an open world.
The notion of separatedness degrees of fuzzy subsets is introduced in fuzzy topological spaces by means of fuzzy closure operators. Later intuitionistic fuzzy rg connectedness was introduced by thakur and rekka chaturvedi in 2006. Definition a topological space x is said to be connected if the empty set. The notion of generalized closed sets in ideal topological spaces was studied by dontchev et. The property we want to maintain in a topological space is that of nearness. The tag wiki for connectedness says, that it is intended for questions concerning pathconnectedness, too. A subset of xis g rconnected if it is g rconnected as a subspace.
X which are both open and closed are the sets x and remark 15. On the other hand, in fields without a formally defined notion of connectivity, the word may be used as a synonym for connectedness. Suppose y is equipped with a topology such that f is. The classes of these sets in a topological space contain the class of open sets.
The notion of connectedness is a basic, useful and fundamental notion in topological spaces. As usual, the approach to connectedness in the topological space set. Generalized semi connectedness in intuitionistic fuzzy. Arc connectedness edit a space x is said to be arcconnected or arcwise connected if any two distinct points can be joined by an arc, that is a path. This applies, for example, to the definitions of interior, closure, and frontier in pseudometric spaces, so these definitions can also be carried over verbatim to a topological space.
Connectedness and continuity in digital spaces with the khalimsky. Connectedness is a topological property quite different from any property we considered in chapters 14. Dec 20, 2012 instead, we need to keep one element of structure we used for graphs before we can talk about connectedness again. On g r connectedness and g r compactness in topological.
Connectedness is a topological property quite different from any. Use features like bookmarks, note taking and highlighting while reading topology for beginners. Let y be a set with topologies t 0 and t 1, and suppose id y. Metric characterization of connectedness for topological.
Any normed vector space can be made into a metric space in a natural way. A topological space is path connected if there is a path between any two of its points, as in the following figure. Dec 31, 2015 the notion of connectedness is a basic, useful and fundamental notion in topological spaces. Connectedness is the sort of topological property that students love. Chapter 6 compactness and connectedness of semi generalized bopen sets in topological spaces 6.
College of arts and science autonomous, pudukkottai 6 22 422, tamilnadu, india. Download it once and read it on your kindle device, pc, phones or tablets. Many mathematicians studied semi connected spaces rigorously. Many researchers have investigated the basic properties of connectedness. Topology underlies all of analysis, and especially certain large spaces such as the dual of l1z lead to topologies that cannot be described by metrics. Recently many fuzzy topological concepts such as fuzzy connectedness have been generalized in intuitionistic fuzzy topological spaces. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. In compact metric spaces uniform connectedness and connectedness are wellknown to coincide, thus the apparent conceptual difference between the two notions disappears. Connectedness is one of the topological properties that is used to distinguish topological spaces.
Connectedness, path connectedness, and uniform connectedness are wellknown concepts. They also initiated the concept of generalized neutrosophic connected and of their properties. Some related but stronger conditions are path connected, simply connected, and nconnected. A subset of a topological space x is a connected set if it is a connected space when viewed as a subspace of x. Topological spaces have been generalized in different ways, for example, the ab stract space 1, the supratopological space. Pdf some strong forms of connectedness in topological spaces.
An example of such spaces is the closure of the graph of sin1x. Connectedness and some concepts related to connectedness of a topological space showing 14 of 49 pages in this thesis. Topological spaces form the broadest regime in which the notion of a continuous function makes sense. The following properties, well known for normed linear spaces of dimension. Neutrosophic complex connectedness in neutrosophic. Box 1664 al khobar 31952, kingdom of saudi arabia abstract. On convergence in lvalued fuzzy topological spaces ramadan, a. Generalizations of some ordinary and extreme connectedness. The aim of this paper is to show the existence of various types of igpr connectedness in the intuitionistic topological spaces. A topological space x is said to be totally disconnected if every subset of x having more than one point is disconnected under the subspace topology theorems about connectedness edit if x \displaystyle x and y \displaystyle y are homeomorphic spaces and if x \displaystyle x is connected, then y \displaystyle y is also connected. In this chapter we introduce the idea of connectedness. A topological space x is said to be g rconnected if x cannot be expressed as a disjoint of two non empty g ropen sets in x.
The resulting connectedness criterion is shown to be valid for all topological spaces, thus completely eradicating the conceptual difference between connectedness and path connectedness. More generally, a set is a metric space if there is a distance defined for it. A new characterization of compact sets in fuzzy number spaces huang, huan and wu, congxin, abstract and applied analysis, 20. Monirul islam 2 1,2 department of mathematics university of gour banga p.
A metric space is a set x where we have a notion of distance. The proof of the following simple conditions for connectedness will be left to the reader. Conceptually speaking there is a substantial difference between connectedness and the other two notions, namely connectedness is defined as the absence of disconnectedness, while path connectedness and uniform connectedness are defined in terms of connecting paths and connecting walks, respectively. We will follow munkres for the whole course, with some occassional added topics or di erent perspectives. Connectedness in lfuzzy topological spaces sciencedirect. More connectedness in topological spaces shyamapada modak 1 and md. A connected space need not\ have any of the other topological properties we have discussed so far. T 1, we say that t 1 is ner than t 0 and that t 0 is coarser than t 1. If v,k k is a normed vector space, then the condition du,v ku. Connectedness and some concepts related to connectedness. For that reason, this lecture is longer than usual. Here is a typical way these connectedness ideas are used.
Stronger and weaker forms of connectedness have been introduced and investigated by noiri. Metric characterisation of connectedness for topological. Mugarjee and roy 2007 discussed a new type of compactness via grills. The following conditions are equivalent for any bito. X y be a surjective mapping from a connected topological space onto a set y. In a fuzzy topological space, sostak introduced a notion of connectedness degree by means of the level topological spaces 20, 21, that is, it can be viewed as connectivity in a topological space. A topological space x,t is said to be disconnected if there exist disjoint nonempty subsets a, b.
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