In 1990 jelic introduced 1, 2locally closed sets and 1, 2 lccontinuity in bitopological space. Balachandran et al 1 introduced the concept of generalized locally closed sets. If f is weakly lc continuous at every ointp of x, then it is. A is an accumulation point of a, then continuity of f at c is equivalent to the condition that lim x. Finally in section 5 we study complete metric spaces. Pdf a new class of locally closed sets and locally closed. On decomposition of generalized continuity article. A note on some unified types of open and locally closed sets. If g is continuous at a and f is continuous at g a, then fog is continuous at a. Thereafter, intensive research on the field of locally closed sets and lc continuous functions has been done as theory developed by balachandran et al. In 1989, ganster and reilly defined lccontinuous functions by using locally closed sets due to bourbaki. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sublc. Introduction in topological spaces locally closed sets were studied more by bourbaki 2 in 1966, which is the intersection of an open set and a closed set.
Locally closed sets and lccontinuous functions 419 we denote the collection of all locally closed subsets of x,t by lcx,t. X, x, is lccontinuous if the inverse with respect to f of any open set in y is closed in x. The study of locally closed sets was introduced by bourbaki 3 in 1966 then. In 1989, ganster and reilly continued the study of locally closed sets and also introduced the concept of lc continuous functions to find a decomposition of continuous functions. Locally closed set, lccontinuous, nearly continuous. X, keywords lc sets, lc continuity, lc irresoluteness. Lcontinuous functions and discussed some of their properties. Reilly, locally closed sets and lccontinuous functions, internat. Feebly locally closed sets and feebly lccontinuous functions, to appear. In this paper, we introduce and investigate the concept of strongly minimal generalized locally closed set in minimal space and biminimal space and the classes of s mglccontinuous maps and smglcirresolute maps, smglc continuous maps and. In this paper, we define a new set called supra bt locally closed and also define supra bt locally continuous functions and investigated some of. A subset a of x, is called supra bt locally closed set, if a u. By using the concept of a locally closed set ganster and.
Decomposition of generalized closed sets in supra topological. Balachandran et al 1 introduced the concept of generalized locally. Stronglyequationlocally closed sets and decompositions. Stone 10 used the term lc for a locally closed set. Let x be an initial universe and let e be a set of parameters.
This enabled them to produce a decomposition of continuity for functions between arbitrary topological spaces. The study of semi open sets and semi continuity in. This enabled them to produce a decomposition of continuity for functions. The nal method, of decomposing a function into simple continuous functions, is the simplest, but requires that you have a set of basic continuous functions to start with somewhat akin to using limit rules to nd limits.
Strongly minimal generalized locally closed sets and locally. Decomposition of closed sets in supra topological spaces. If a subset s of x is m xlocally closed, then it is strongly minimal. In this paper we introduce and study three different notions of generalized continuity, namely lcirresoluteness, lccontinuity and sublccontinuity. Decomposition of j closed sets in bigeneralized topo.
Asubsetaof, 1,2 is,locallyclosedif andonlyifa is the union of a 1,2open set and a 1,2 closed set. T be a locally compact hausdor second countable space lccb. Lccontinuous if the inverse with respect to f of any open set in y is locally closed in x. Here, is, locally closed but not, locally closed set. Strongly minimal generalized locally closed sets 2649 the case of b being a m xclosed set, we have, a. Locally closed sets and lccontinuous functions hindawi. The concept of locally closed sets is due to kuratowski and sierpinski 10, in 1921. For any f 2 l1 and any 0 there is a g in ccrn such that jf gjdm sets fx2xjfx cgare both open sets in x. In this paper we introduce the notion of locally closed sets which are denoted by lc sets and study some of the fundamental properties of lc sets in generalized topological spaces.
Then in 1989 ganster and reilly used locally closed sets to define lccontinuity in a topological space. On a type of decomposition of continuity springerlink. Locally closed sets and lc continuous functions 419 we denote the collection of all locally closed subsets of x,t by lc x,t. Baire category theorem is proved and, as an application, it is shown that continuous, nowhere di erentiable functions form a set of second category in the space of continuous functions. Locally closed sets and lccontinuous functions article pdf available in international journal of mathematics and mathematical sciences 123 january 1989 with 633. In this paper we consider a stronger form of lccontinuity called continue reading. Kuratowski 4 was introduced the local function in ideal spaces. We introduce the notion of, locally closed continuous functions on generalized topological space and investigate some of their characterizations. He showed that the hereditarily absolute fg spaces coincide with the hereditarily locally compact spaces for hausdorff spaces. On decomposition of gr closed set in topological spaces. V, where u is supra bt open in x, and v is supra bt closed in x. Nagaveni2 1,2 department of mathematics, coimbatore institute of technology, india abstract. Asubsetaof, 1,2 is,locallyclosedif andonlyifa is the union of a 1,2open set and a 1,2closed set.
On locally deltageneralized closed sets and ldeltagc continuous functions. And of course we therefore also have the parallel characterization in terms of closed sets. A subset s of a topological space x is locally closed if it is the intersection of an open and a closed set. Ganster and reilly used locally closed sets in and 14 to define the concept of lccontinuity, i. Let px denote the power set of x and let a be a nonempty subset of e. Decomposition of closed sets in supra topological spaces s. Y is said to eb weakly lc continuous at x 2 x if for ache open set v of y ontainingc f x, there exists a loalcly closed set u in x ontainingc x such that f u v. Full text of on some locally closed sets and spaces in ideal.
In 3, the notions of g open sets, g semiopen sets, g pre open sets and g open sets in generalized topological spaces are analyzed. A new class of locally closed sets and locally closed. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The purpose of our note is to show that every subset of any topological space is the intersection of a preopen set and a preclosed set, hence. Stronglyequationlocally closed sets and decompositions of. In other words, a soft set over x is a parameterized family of subsets of the universe x. Continuous functions do not necessarily map closed sets to closed sets. In this paper, the notions of locally \\mu \closed sets, \\mu \\t\sets, \\mu \\b\sets have been introduced. The aim of this paper is to introduce and study the classes of g locally closed set and different notions of generalization of continuous functions namely g. Strongly minimal generalized locally closed sets and. A subset a of a topological space x is strongly minimal glc if and only if there exists a smg open set p such that ap.
Locally closed sets and lccontinuous functions article pdf available in international journal of mathematics and mathematical sciences 123 january 1989 with 633 reads how we measure reads. Stone 27 has used the term fg for a locally closed subset. In 1989, ganster and reilly defined lc continuous functions by using locally closed sets due to bourbaki. B where a is strongly mgopen in x and b is strongly mgclosed in x. This lemma, variously attributed to arzela, escoli, montel, vitali, and so on, is of importance. Stone has used the term fg for a locally closed subset. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sub lc continuous and nearly continuous in the sense of ptak. We rst notice that the collection c of all open sets with compact closure form a basis for t. Ganster and reilly used locally closed sets in 4 to define lccontinuity and lcirresoluteness. In this paper, we introduce a new class of sets called supra locally closed sets and new class of maps called supra locally continuous functions. Some weaker forms of decompositions in supra topological.
Some generalizations of locally closed sets iranian journal of. Also we introduce the concept of, locally closed continuous functions on generalized. Y is said to eb weakly lc continuous at x 2 x if for ache open set v of y ontainingc f. Scott, can we say is that if f is continuous and injective, then f maps closed sets to closed sets.
R is continuous if and only if, for every c2r the sets fx2xjfx cgand fx2xjfx 5 cgare both closed sets in x. Balachandran 22 introduced the concept of generalized locally closed sets and obtained di. Rwclosed sets, rwopen sets, locally closed sets, rwlocally closed sets 1. Continuous functions do not necessarily map closed sets to. Extensive research on locally closedness and generalizing locally closedness were done in recent years. A pair f, a is called a soft set over x, where f is a mapping given by f. All three notions are defined by using the concept of a locally closed set. Since nite unions of open sets are open, it follows that. Btlocally closed sets and btlocally continuous functions in supra topological spaces. Example last day we saw that if fx is a polynomial, then fis continuous at afor any real number asince lim x. In this paper we introduce and study three different notions of generalized continuity, namely lc irresoluteness, lc continuity and sub lc continuity. In 1989 ganster and reilly 6 introduced and studied the notion of lccontinuous functions via the concept of locally closed sets. Ganster and reilly used locally closed sets in 4 to define lc continuity and lc irresoluteness. New types of transitive functions and minimal systems.
Research article some properties of 1,2 locally closed sets. Some weaker forms of decompositions in supra topological spaces. Pdf a new class of locally closed sets and locally. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sublccontinuous and nearly continuous in the sense of ptak.
A subset s of x is called strongly minimal generalized locally closed set smg. According to them a function is said to be lccontinuous if the inverse image of every open set in is locally closed set in. How can a continuous function map closed sets to open sets. Ac denotes the complement of a in x unless explicitly stated. We discuss some properties of these functions and show that. I2 introduced and investigated some properties of regular generalized locally closed sets and rglcontinuous functions. Aug 29, 20 on various decompositions of continuous and some weakly continuous functions. In addition, an explicit example of such a function on lp 1 continuous, nowhere locally uniformly. The family of all loalcly closed sets of x ontainingc a oinpt x 2 x is denoted by lc x.
Sundaram introduced sg locally closed sets, semi locally closed, locally semi closed and investigated some of their topological properties. In 1989, ganster and reilly continued the study of locally closed sets and also introduced the concept of lccontinuous functions to find a decomposition of continuous functions. A study on fuzzy locally b closed sets 50 fuzzy locally b closed compact spaces are introduced and some of their charecterizations and properties are established. In a recent paper, gnanambal and balachandran 1 introduced the classes of. Locally closed sets and lccontinuous functions core. Introduction kuratowski and sierpinski 11 introduced the notion of locally closed sets in topological 6, a subset of a topological space x. Full text of on some locally closed sets and spaces in. Kelly8 initiated the notion of bitopological spaces. Compact sets of functions and function rings david gale a widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. A study on fuzzy locally bclosed sets 50 fuzzy locally bclosed compact spaces are introduced and some of their charecterizations and properties are established. Density of continuous functions in l1 math problems and.
Later on, jelic 9 extended their results to the bitopological setting. Introduction kuratowski and sierpinski 5 considered the difference of. On decomposition of g closed sets in topological spaces. The inversetrigonometric functions, in their respective i. I2 introduced and investigated some properties of regular generalized locally closed sets and rgl continuous functions. Stone used the term fg for a locally closed subsets.
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