Topological space, in mathematics, generalization of euclidean spaces in which the idea of closeness, or limits, is described in terms of relationships between sets rather than in terms of distance. The particular distance function must satisfy the following conditions. In this paper we consider, discuss, improve and generalize recent fixed point results for mappings in bmetric, rectangular metric and brectangular metric spaces established by dukic et al. This theorem implies that the completion of a metric space is unique up to isomorphisms. A metric space is a set x where we have a notion of distance. X r, we say that the pair m x, d is a metric space if and only if d satisfies the following. Universal property of completion of a metric space let x. Real analysismetric spaces wikibooks, open books for an. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are.
The ideas of convergence and continuity introduced in the last sections are useful in a more general context. Every metric space can be isometrically embedded in a complete metric space ii. For modern metric system, see international system of units. Now we present the definition of cauchy sequence, convergent sequence and complete bmetric space. Y into a complete metric space y and any completion x. This is a basic introduction to the idea of a metric space. Set theory and metric spaces kaplansky chelsea publishing company 2nd. Eclasses, which we now call metric spaces, and vclasses,15 a metric space with a weak version of the triangle inequality, were less general, but easier to work with. Bidholi, via prem nagar, dehradun uttarakhand, india.
First course in metric spaces presents a systematic and rigorous treatment of the subject of metric spaces which are mathematical objects equipped with the notion of distance. Note that iff if then so thus on the other hand, let. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex. A of open sets is called an open cover of x if every x. In calculus on r, a fundamental role is played by those subsets of r which are intervals. A metric space is called complete if every cauchy sequence converges to a limit.
An introduction to metric spaces and fixed point theory includes an extensive bibliography and an appendix which provides a complete summary of the concepts of set theory, including zorns lemma, tychonoffs theorem, zermelos theorem, and transfinite induction. In some cases, when the contractive condition is of nonlinear type, the above strategy cannot be used. A metric space is a set xtogether with a metric don it, and we will use the notation x. Turns out, these three definitions are essentially equivalent. The general idea of metric space appeared in fr echet 1906, and metricspace structures on vector spaces, especially spaces of functions, was developed by fr echet 1928 and hausdor 1931. Also, we prove a geraghty type theorem in the setting of bmetric spaces as well as a boydwong type theorem in the framework of b. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. We just saw that the metric space k 1 isometrically embeds into 2 k in fact, a stronger result can be shown. This volume provides a complete introduction to metric space theory for undergraduates.
However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis. The abstract concepts of metric spaces are often perceived as difficult. An open neighbourhood of a point p is the set of all points within of it. I introduce the idea of a metric and a metric space framed within the context of rn. It is well known that a metric space is compact if and only if it is complete and totally bounded see, e. It covers the topology of metric spaces, continuity, connectedness, compactness and product spaces, and includes results such as the tietzeurysohn extension theorem, picards theorem on ordinary differential equations, and the set of discontinuities of the pointwise limit of a sequence of continuous functions. The space c a, b of continuous realvalued functions on a closed and bounded interval is a banach space, and so a complete metric space, with respect to the supremum norm.
Chapter 9 the topology of metric spaces uci mathematics. However, the supremum norm does not give a norm on the space c a, b of continuous functions on a, b, for it may contain unbounded functions. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. In r2, draw a picture of the open ball of radius 1 around the origin.
Informally, 3 and 4 say, respectively, that cis closed under. Often, if the metric dis clear from context, we will simply denote the metric space x. Pdf various generalizations of metric spaces and fixed. Thus, rst, the only point yat distance 0 from a point xis y xitself. That is, dx, y is the sum of the euclidean distances of x and y from the origin.
A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. The basic idea that we need to talk about convergence is to find a. The analogues of open intervals in general metric spaces are the following. Metric spaces, completeness completions baire category theorem 1. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. A metric space is just a set x equipped with a function d of two variables which measures the distance between points. But this follows from the corollary in the preceding section when u x. The following properties of a metric space are equivalent.
In particular we will be able to apply them to sequences of functions. This book is a step towards the preparation for the study of more advanced topics in analysis such as topology. On some fixed point results in bmetric, rectangular and b. Nov 22, 2012 we discuss the introduced concept of g metric spaces and the fixed point existing results of contractive mappings defined on such spaces. It is also sometimes called a distance function or simply a distance often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used we already know a few examples of metric spaces.
A metric space consists of a set x together with a function d. Metric space, in mathematics, especially topology, an abstract set with a distance function, called a metric, that specifies a nonnegative distance between any two of its points in such a way that the following properties hold. This article is about the development and the history of the standards used in the metric system. An introduction to metric spaces and fixed point theory.
For more details about the linear case, we refer the reader to. Completion of metric spaces explanation of the proof. All spaces satisfying the first axiom of countability, and only they, are the images of metric spaces under continuous open. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. A point p is a limit point of the set e if every neighbourhood of p contains a point q. Metricandtopologicalspaces university of cambridge. A metric space consists of a set xtogether with a function d. Ais a family of sets in cindexed by some index set a,then a o c. We introduce metric spaces and give some examples in section 1. Case ii and are in the different ray from the origin. Let x,d be a metric space and let s be a subset of x, which is a metric space in its own right. Ne a metric space is a mathematical object in which the distance between two points is meaningful. A topological space which is the image of a metric space under a continuous open and closed mapping is itself homeomorphic to a metric space. Metric spaces constitute an important class of topological spaces.
A subspace of a metric space always refers to a subset endowed with the induced metric. A good book for metric spaces specifically would be o searcoids metric spaces. Remarks on g metric spaces and fixed point theorems fixed. You have met or you will meet the concept of a normed vector space both in algebra and analysis courses.
A comprehensive, basic level introduction to metric spaces and fixed point theory an introduction to metric spaces and fixed point theory presents a highly selfcontained treatment of the subject that is accessible for students and researchers from diverse mathematical backgrounds, including those who may have had little training in mathematics beyond calculus. The general idea of metric space appeared in fr echet 1906, and metric space structures on vector spaces, especially spaces of functions, was developed by fr echet 1928 and hausdor 1931. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. However, under continuous open mappings, metrizability is not always preserved. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of. Xthe number dx,y gives us the distance between them. Pdf this chapter will introduce the reader to the concept of metrics a class of functions which is. Jan 22, 2012 this is a basic introduction to the idea of a metric space.
The most familiar is the real numbers with the usual absolute value. A metric space is a pair x, d, where x is a set and d is a metric on x. Chapter 1 metric spaces islamic university of gaza. A metric space is a set x that has a notion of the distance dx, y between every. Remarks on g metric spaces and fixed point theorems. Theorem in a any metric space arbitrary intersections and finite unions of closed sets are closed. Every metric space can be isometrically embedded in a complete metric space i. Introduction when we consider properties of a reasonable function, probably the. A good book for real analysis would be kolmogorov and fomins introductory real analysis. Neighbourhoods and open sets in metric spaces although it will not be clear for a little while, the next definition represents the first stage of the generalisation from metric to topological spaces. In particular, we show that the most obtained fixed point theorems on such spaces can be deduced immediately from fixed point theorems on metric or quasi metric spaces. What topological spaces can do that metric spaces cannot82 12. In mathematics, a metric space is a set together with a metric on the set. U nofthem, the cartesian product of u with itself n times.
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