You have met or you will meet the concept of a normed vector space both in algebra and analysis courses. In some cases, when the contractive condition is of nonlinear type, the above strategy cannot be used. Theorem in a any metric space arbitrary intersections and finite unions of closed sets are closed. What topological spaces can do that metric spaces cannot82 12. 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. A metric space is a set x where we have a notion of distance. All spaces satisfying the first axiom of countability, and only they, are the images of metric spaces under continuous open. Ne a metric space is a mathematical object in which the distance between two points is meaningful. Informally, 3 and 4 say, respectively, that cis closed under.
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. Every metric space can be isometrically embedded in a complete metric space ii. But this follows from the corollary in the preceding section when u x. Jan 22, 2012 this is a basic introduction to the idea of a metric space. 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 calculus on r, a fundamental role is played by those subsets of r which are intervals. We just saw that the metric space k 1 isometrically embeds into 2 k in fact, a stronger result can be shown. Let x,d be a metric space and let s be a subset of x, which is a metric space in its own right. Chapter 9 the topology of metric spaces uci mathematics.
A metric space is called complete if every cauchy sequence converges to a limit. The particular distance function must satisfy the following conditions. Xthe number dx,y gives us the distance between them. Introduction when we consider properties of a reasonable function, probably the. A subspace of a metric space always refers to a subset endowed with the induced metric. Y into a complete metric space y and any completion x.
This is a basic introduction to the idea of a metric space. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. 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. Ais a family of sets in cindexed by some index set a,then a o c. 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. It is well known that a metric space is compact if and only if it is complete and totally bounded see, e. 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. Chapter 1 metric spaces islamic university of gaza. A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. A of open sets is called an open cover of x if every x. This article is about the development and the history of the standards used in the metric system. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex.
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. The basic idea that we need to talk about convergence is to find a. 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. U nofthem, the cartesian product of u with itself n times.
Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. Bidholi, via prem nagar, dehradun uttarakhand, india. A metric space consists of a set xtogether with a function d. 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. The ideas of convergence and continuity introduced in the last sections are useful in a more general context. 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. On some fixed point results in bmetric, rectangular and b. The following properties of a metric space are equivalent. Case ii and are in the different ray from the origin. Now we present the definition of cauchy sequence, convergent sequence and complete bmetric space. Real analysismetric spaces wikibooks, open books for an. 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.
X r, we say that the pair m x, d is a metric space if and only if d satisfies the following. 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. Metricandtopologicalspaces university of cambridge. 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. A metric space is a pair x, d, where x is a set and d is a metric on x. That is, dx, y is the sum of the euclidean distances of x and y from the origin. The analogues of open intervals in general metric spaces are the following. The most familiar is the real numbers with the usual absolute value. We introduce metric spaces and give some examples in section 1. Turns out, these three definitions are essentially equivalent. Pdf various generalizations of metric spaces and fixed.
An introduction to metric spaces and fixed point theory. Completion of metric spaces explanation of the proof. A metric space is a set x that has a notion of the distance dx, y between every. A good book for real analysis would be kolmogorov and fomins introductory real analysis. This book is a step towards the preparation for the study of more advanced topics in analysis such as topology. 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. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. 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. Often, if the metric dis clear from context, we will simply denote the metric space x. I introduce the idea of a metric and a metric space framed within the context of rn. Remarks on g metric spaces and fixed point theorems fixed. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc.
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. A good book for metric spaces specifically would be o searcoids metric spaces. Set theory and metric spaces kaplansky chelsea publishing company 2nd. For more details about the linear case, we refer the reader to. The abstract concepts of metric spaces are often perceived as difficult. Remarks on g metric spaces and fixed point theorems. An open neighbourhood of a point p is the set of all points within of it. A metric space consists of a set x together with a function d. This volume provides a complete introduction to metric space theory for undergraduates.
Metric spaces, completeness completions baire category theorem 1. Every metric space can be isometrically embedded in a complete metric space i. Universal property of completion of a metric space let x. 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. 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. 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 a set xtogether with a metric don it, and we will use the notation x. In particular we will be able to apply them to sequences of functions. 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.
Thus, rst, the only point yat distance 0 from a point xis y xitself. Now, t satisfies cirics contractive condition in the complete metric space x. A metric space is just a set x equipped with a function d of two variables which measures the distance between points. For modern metric system, see international system of units. Note that iff if then so thus on the other hand, let. Metric spaces constitute an important class of topological spaces. 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. In mathematics, a metric space is a set together with a metric on the set. Pdf this chapter will introduce the reader to the concept of metrics a class of functions which is. This theorem implies that the completion of a metric space is unique up to isomorphisms. 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. A point p is a limit point of the set e if every neighbourhood of p contains a point q. In r2, draw a picture of the open ball of radius 1 around the origin. However, under continuous open mappings, metrizability is not always preserved.
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