We then looked at some of the most basic definitions and properties of pseudometric spaces. Generalized topological spaces with associating function. Topology is really just the study of continuous functions. Slightly wbcontinuous functions in topological spaces. Second, we allow for the possibility that the whole space is not open. In this paper we present new definition and give two examples which is clarified in this diagram of suprapre closed sets maps, for that we support it with a theorem keywords. Suppose h is a subset of x such that f h is closed where h denotes the closure of h. It is assumed that measure theory and metric spaces are already known to the reader. Throughout the present paper and or simply and denote topological spaces on. Strong forms of stronger and weaker forms of continuous map have been in troduced and investigated by several mathematicians. A rigorous introduction to set theory, topological spaces, continuity, separation, metrizability, compactness, connectedness, function spaces, and algebraic topology kindle edition by warner, steve. Any function from a discrete space to any other topological space is continuous.
Y is continuous if v is open in y f 1v is open in x. On slight omega continuity and irresoluteness between generalized topological spaces author. Essentially, topological spaces have the minimum necessary structure to allow a definition of continuity. X y, where x and y are topological spaces is continuous if it is continuous at each point. Chapter 5 functions on metric spaces and continuity.
Well give some examples and define continuity on metric spaces, then show how continuity can be stated without reference to metrics. Recently, jafari and noiri 5 introduced new generalization of contra continuity called contra continuity. Maria singam, on contra bcontinuous functions in topological spaces, international journal of mathematical archieve512, 2014, 6674. We take functions defined on all of topological space x, so there.
X y, where x and y are topological spaces, is continuous at p if whenever u is an open set in y containing fp, there is an open set v in x containing p with fv. Recall from singlevariable calculus that a function f. In 3, dontchev introduced and investigated a new notion of continuity called contra continuity. Pdf sgcontinuity in topological spaces researchgate. If you liked what you read, please click on the share button. Any function from any topological space to an indiscrete space is continuous. Ekici 3 introduced and studied bcontinuous functions in topological spaces. Continuity for topological spaces mathematics stack exchange.
Topologycontinuity and homeomorphisms wikibooks, open. The preimage under such a function of any set containing the constant value is the whole. Parimelazhagan, gbhomeomorphisms and contra gbcontinuous maps in topological spaces, international journal of computer applications, vol. Introduction to topological spaces and setvalued maps. Continuity in almost any other context can be reduced to this definition by an appropriate choice of topology. Any constant function is continuous regardless of the topologies on the two spaces. X y from a topological space x to a topological space y is said to be. Functions between topological spaces being continuous at a. 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. Download it once and read it on your kindle device, pc, phones or tablets. Hence, some points of the universe may be beyond any open set. It will be convenient to restate continuity in terms of continuity at a point.
The objective of the paper is to introduce a new types of continuous maps and irresolute functions called. Ekici 3 introduced and studied b continuous functions in topological spaces. Closed sets, hausdorff spaces, and closure of a set. In 1 al abdullah and abed introduced a new class of sets in a topological space, known as bc open paper, we employ this notion of sets to introduce and investigate contra continuous functions, called contra bc continuous functions. About the continuity of a function between topological spaces. In 3, dontchev introduced and investigated a new notion of continuity called contracontinuity. Chapter 5 functions on metric spaces and continuity when we studied realvalued functions of a real variable in calculus, the techniques and theory built on properties of continuity, differentiability, and integrability. Pdf contra continuous functions in topological spaces.
X y, where x and y are topological spaces is continuous if. Cauchy spaces provide a general setting for studying completions. Solomon lefschetz in order to forge a language of continuity, we begin with familiar examples. Some stronger forms of g pre continuous functions in topology 1.
This material will motivate the definition of topology in chapter 2 of your textbook. These lack some of the conveniences of banach spaces, but their expressions as limits of banach spaces is often su cient. First, these families of subsets are not closed under intersections. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Cauchy spaces axiomatize the ability to test whether a net is cauchy. Functions between topological spaces being continuous at a point.
We say that f is a topological isomorphism or a homeomorphism if f is a bijection and both f and f 1 are continuous. What does continuity of a function mapping a topological space to a real line interval mean. Contra bc continuous functions in topological spaces. Slightly b continuous functions in topological spaces raja mohammad latif 1, muhammad rafiq raja 2, muhammad razaq 3 1department of mathematics and natural sciences prince mohammad bin fahd university, al khobar, sa udi arabia 2department of mathematics, university of sargodha, mandi bahauddin campus, pakistan 3department of mathematics, postgraduate horizon cll ege, chakwal. In this chapter, we want to look at functions on metric. Paper 1, section ii 12e metric and topological spaces.
A new weaker form of the closed sets called gclosed sets is introduced and investigated by sudha et. Use features like bookmarks, note taking and highlighting while reading topology for beginners. Lesson 8 continuity in r and c lesson 9 topological spaces lesson 10 separation and countability lesson 11 metrizable spaces lesson 12 compactness lesson continuity and homeomorphisms lesson 14 connectedness lesson 15 function spaces lesson 16. In this paper we assume that such points are associated. We introduce slightly 1,2 continuous and slightly 1,2irresolute functions as two new classes of functions on generalized topological spaces. A topological space in which the points are functions is called a function space. Recently, jafari and noiri 5 introduced new generalization of contracontinuity called contra continuity. Pdf abstract in this paper we introduce some new class of functions called. On slight omega continuity and irresoluteness between.
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