Almost homeomorphisms on bigeneralized topological spaces 1855 let x. Jan 15, 2018 homeomorphism between topological spaces this video is the brief definition of a function to be homeomorphic in a topological space and in this video the main conditions are mentioned to be. Further some of its properties and characterizations are established. Introduction to generalized topological spaces zvina. This avoids the need to worry about inverse functions. Keywords gopen map, g homeomorphism, gchomeomorphisms definition 2. On generalized topology and minimal structure spaces. Mashhour et al 6 introduced the supra topological spaces and studied, continuous functions and s continuous functions. Nhomeomorphism and nhomeomorphism in supra topological spaces l. Generalized homeomorphism in vague topological spaces. Unlike in algebra where the inverse of a bijective homomorphism is always a homomorphism this does not hold for.
A topological space x is homeomorphic to a space y if there exists a. If a property of a space applies to all homeomorphic spaces to, it is called a topological property. K is expansive if there exists e 0 such that if x,y e k satisfy dfx,fy x such that fx y. X if for each g yopen set v containing fx there exists a g xopen set u containing. Nhomeomorphism and n homeomorphism in supra topological spaces. Two spaces are called topologically equivalent if there exists a homeomorphism between them. We combine the preceding two lemmas to get the following. Section 8 outlines the contributions of the author.
In general, the nonhomeomorphism of two topological spaces is proved by specifying a topological property displayed by only one of them compactness, connectedness, etc e. Many researchers have generalized the notions of homeomorphism in topological spaces. This implies an equivalence relation on the set of topological spaces verify that the reflexive, symmetric, and transitive properties are implied by the homeomorphism. Section 6 and section 7 marches on with the insights of ideal and fuzzy topology. If a homeomorphism exists between two spaces, the spaces are said to be homeomorphic. In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. A metric space is a set x where we have a notion of distance. Keywords gopen map, ghomeomorphism, gchomeomorphisms definition 2. Show that x is a t 1space if and only if each point of x is a closed set. The closure of a and the interior of a with respect to. X y is said to be generalized continuous if f 1v is g open in x for each set v of y definition 8 2.
T1, soft generalized hausdorff, soft generalized regular, soft generalized normal and soft generalized completely regular spaces in soft generalized. Nano generalized pre homeomorphisms in nano topological space. N levine6 introduced the concept of generalized closed sets and the class of continuous function using gopen set semi open sets. Rajarubi abstract in this paper, we introduce a new class of sets called. Lecture notes on topology for mat35004500 following jr.
Thus topological spaces and continuous maps between them form a category, the category of topological spaces. A map may be bijective and continuous, but not a homeomorphism. Malghan 3 introduced the concept of generalized closed maps in topological spaces. For any internal point x of x its image lie in some subset of y homeomorphic to r n so f x cannot lie in. Homeomorphisms on topological spaces examples 1 mathonline. It means that, for every a g s, the restriction map h h\a, h g g, is onto. Combining this with the bijection between topologies and closure. Examples are connectedness, compactness, and, for a plane domain, the number of components of the boundary. Topologycontinuity and homeomorphisms wikibooks, open. Boonpok boonpok 4 introduced the concept of bigeneralized topological spaces and studied m,nclosed sets and m,nopen sets in bigeneralized topologicalspaces. The most general type of objects for which homeomorphisms can be defined are topological spaces. Also intuitionistic generalized preregular homeomorphism and intuitionistic generalized preregular homeomorphism were introduced and.
If both and are continuous, then is called a homeomorphism. Generalized closed sets in topological spaces in this section, we introduce the concept of. Also we introduce the new class of maps, namely rgw. Example 4 5 interval homeomorphisms any open interval of is homeomorphic to any other open interval. To handle this, and many other more general examples, one can use a more general concept. In general, weak homeomorphisms do not preserve baire spaces.
Weprovide some examples of gtspaces and study key topological notions continuity, separation axioms, cardinal invariants in terms of. More on generalized homeomorphisms in topological spaces emis. Moreover, he studied the simplest separation axioms for generalized topologies in 2. An equivalent way to define homeomorphism is as a bijective, continuous, open map maps open sets to open sets. For various classes k of topological spaces we prove that if x\. If there is a ghomeomorphism between x and y they are said to be ghomeomorphic denoted by x. In this paper, we introduce and study a new class of maps called generalized open maps and the notion of generalized homeomorphism and gc homeomorphism in topological spaces. Introduction the concept of the closed sets in topological spaces has been. Y, 1 is called generalized db homeomorphism briefly g db homeomorphism, if both f and. If a subset, can be mapped to another, via a nonsingular linear. The bijective mapping f is called a ghomeomorphism from x to y if both f and f.
X y is called gcontinuous on x if for any gopen set o in y, f. Almost homeomorphisms on bigeneralized topological spaces. This kind of homeomorphism can be generalized substantially using linear algebra. The notion homeomorphism plays a very important role in topology. Pdf homeomorphism on intuitionistic topological spaces. Sivakamasundari 2 1 departmen t of mathematics,kumaraguru college of technology, coimbatore,tamilnadu meena. We obtain several characterizations and properties of almost. Homeomorphisms on topological spaces examples 2 mathonline. Homeomorphismtopological spaces version of cantorbernsteinschroeder theorem. Generalized homeomorphism in topological spaces call for paper june 2020 edition ijca solicits original research papers for the june 2020 edition. Devi et al 5 defined and studied generalized semi homeomorphism and gschomeomorphism in topological spaces. Homework 5, due thursday, october 11, 2012 do any 5 of the 8 problems. Throughout the thesis x, y and z denote topological spaces under simple extension, on which no seperation axioms.
X if for each g yopen set v containing fx there exists a. We will now look at some examples of homeomorphic topological spaces. A topological property is defined to be a property that is preserved under a homeomorphism. Soft generalized separation axioms in soft generalized topological spaces. In this paper we study some other properties of g c homeomorphism and the pasting lemma for g irresolute maps. It is not even neccessary that the two topological spaces have to be defined on the same base space. The \\ mu \ open sets are sets where the closure has been considered in topological space and interior in generalized topological space. Lellis thivagar 4 introduced nano homeomorphisms in nano topological spaces. Vigneshwaran department of mathematics, kongunadu arts and science college, coimbatore,tn,india. Generalized topology of gtspace has the structure of frame and is closed under arbitrary unions and finite intersections modulo small subsets.
Introduction to generalized topological spaces we introduce the notion of generalized topological space gtspace. The family of small subsets of a gtspace forms an ideal that is compatible with the generalized topology. The modified form of these sets and generalized continuity were further developed by many mathematicians 4,5. The word homeomorphism comes from the greek words homoios similar or same and morphe shape, form, introduced to mathematics by henri poincare in 1895. Two topological spaces and are said to be homeomorphic, denoted by, if there exists a homeomorphism between them. In this paper, we introduce the concept of strongly supra ncontinuous function and perfectly. Note that and are each being interpreted here as topological subspaces of. X, y, is said to be generalized minimal homeomorphism briefly g m i homeomorphism if and are gm i continuous maps. General terms 2000 mathematics subject classification. A subset a of a topological space x is said to be closed set in x if clinta contained in u whenever u is gopen definition 2.
Introduction to generalized topological spaces 51 assume that b. Homeomorphisms are the isomorphisms in the category of topological spacesthat is, they are the mappings that preserve all the topological properties of a given space. Soft generalized separation axioms in soft generalized. In 1970, levine generalized the concept of closed sets to generalized closed sets1. Note that every discrete space is a regular topological manifold. To support the definition of gtspace we prove the frame embedding modulo compatible ideal theorem. Homeomorphism is the notion of equality in topology and it is a somewhat relaxed notion. Closedopen maps and homeomorphism are discussed in section 5.
In this paper we study some other properties of g chomeomorphism and the pasting lemma for g irresolute maps. In this paper, we study a new space which consists of a set x, general ized topologyon x and minimal structure on x. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Devi2 have studied generalization of homeomorphisms and.
Y, 1 is called generalized dbhomeomorphism briefly g dbhomeomorphism, if both f and. In general, the non homeomorphism of two topological spaces is proved by specifying a topological property displayed by only one of them compactness, connectedness, etc e. This notion has been studied extensively in recent years by many topologists because generalized closed sets are. Oct 06, 2016 we show that this is not necessarily true in generalized topological spaces. In this paper, we first introduce hgclosed maps in topological spaces. He also introduced the notions of associated interior and closure operators and continuous mappings on generalized neighborhood. If x,g x and y,g y are generalized topological spaces, then a function f. Suppose m is a topological space and m is a point in m. The particular distance function must satisfy the following conditions. Homeomorphism in topological spaces rs wali and vijayalaxmi r patil abstract a bijection f. In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Topological properties preserved by weakly discontinuous maps.
Pdf homeomorphism criteria for the theory of grid generation. Closed sets, hausdorff spaces, and closure of a set. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. T1, soft generalized hausdorff, soft generalized regular, soft generalized normal and soft generalized completely regular spaces in soft generalized topological spaces are defined and studied. The purpose of this paper is to show the existence of open and closed maps in intuitionistic topological spaces. Closed mapping in topology was introduced by malgham 7. By combining the concepts of closedness and gclosedness,julian dontchev 4. A new type of homeomorphism in bitopological spaces. For example, can be mapped to by the continuous mapping. We will now look at some more examples of homeomorphic topological spaces. X y is a homeomorphism between topological spaces x and y. Balachandran1 et al introduced the concept of generalized continuous map in a topological space.