This volume provides a complete introduction to metric space theory for undergraduates. The equivalence between closed and boundedness and compactness is valid in nite dimensional euclidean. I introduce the idea of a metric and a metric space framed within the context of rn. Let x be a topological space and let y,d be a metric space. Generalized nmetric spaces and fixed point theorems. If the subset f of cx,y is totally bounded under the uniform metric corresponding to d, then f is equicontinuous under d. We then have the following fundamental theorem characterizing compact metric spaces. 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. In these, the distance function is defined by a norm. Although strictly speaking the metric space is the pair x, d it is a common practice to refer to x itself as being the metric space, if the metric d is understood from context.
In mathematics, a metric space is a set together with a metric on the set. Lecture 3 complete metric spaces 1 complete metric spaces 1. We begin with the familiar notions of magnitude and distance on the real line. Normed vector spaces and metric spaces were going to develop generalizations of the ideas of length or magnitude and distance. In the next two chapters, we will look at two important special cases of metric spaces, namely normed linear spaces and inner product spaces. Continuity, contractions, and the contraction mapping principle 4 acknowledgments 6 references 6 1. Chapter 1 metric spaces islamic university of gaza. Metric geometry a metric on a set x is a function d. In the case of metric spaces, the compactness, the countable compactness and the sequential compactness are equivalent. Y continuous if and only if, for every cauchy sequence fx igin x converging to x2x, lim i fx i f lim i x i proof.
A of open sets is called an open cover of x if every x. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. It turns out that sets of objects of very different types carry natural metrics. It helps to have a unifying framework for discussing both random variables and stochastic processes, as well as their convergence, and such a framework is provided by metric spaces. These questions are subsumed by computing simulation hemimetrics between 1 and. Gahler 3,4 introduced the concept of 2metric as a possible generalization of usual notion of a metric space.
Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. But as we will see in examples it is often possible to assign different metrics to. Wesaythatasequencex n n2n xisacauchy sequence ifforall0 thereexistsann. The above two nonstandard metric spaces show that \distance in this setting does not mean the usual notion of distance, but rather the \distance as determined by the.
As metric spaces one may consider sets of states, functions and mappings, subsets of euclidean spaces, and hilbert spaces. In addition, goebel and kirk 30 studied some iterative processes for nonexpansive mappings in the hyperbolic metric space, and in 1988, xie. Metricandtopologicalspaces university of cambridge. The most familiar is the real numbers with the usual absolute value. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric. Moreover the concepts of metric subspace, metric superspace, isometry i. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. Also recal the statement of lemma a closed subspace of a complete metric space is complete.
Several fixed point theorems in convex bmetric spaces. Informally, 3 and 4 say, respectively, that cis closed under. Recall that every normed vector space is a metric space, with the metric dx. Introduction when we consider properties of a reasonable function, probably the. There are many ways to make new metric spaces from old. This is a basic introduction to the idea of a metric space. An open set in a metric space may be the union of many such sets. First, suppose f is continuous and let u be open in y. Metric spaces, topological spaces, and compactness 253 given s. By the usual abuse of notation, when only one metric.
In any case, in a metric space, we have the notion of distance between points. If x,d is a metric space and a is a nonempty subset of. X y between metric spaces is continuous if and only if f. 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. Real analysismetric spaces wikibooks, open books for an. N such that dx m,x n metric spaces lecture notes for ma2223 p. Completeness and completion compactness in metric spaces. If the underlying metric space is an open subset of a euclidean space, we obtain a natural chain monomorphism from general metric currents to. Here d is the metric on x, that is, dx, y is regarded as the distance from x to y.
The properties of the banach spaces and metric spaces studied in the ribe program. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. In chapter 2 we learned to take limits of sequences of real numbers. The metric topology on a metric space m is the coarsest topology on m relative to which the metric d is a continuous map from the product of m with itself to the nonnegative real numbers. A metric space is a pair x, d, where x is a set and d is a metric on x. This is an isometry when r is given the usual metric and r2 is given the 2dimensional euclidean metric, but not. Summer 2007 john douglas moore our goal of these notes is to explain a few facts regarding metric spaces not included in the. Metric spaces the limit is often identi ed as the fundamental basis of calculus. Ais a family of sets in cindexed by some index set a,then a o c. Note that iff if then so thus on the other hand, let.
As for the box metric, the taxicab metric can be generalized to rnfor any n. Cambridge core abstract analysis metric spaces by e. This approach leads to the idea of a metric space, first suggested by m. To prove the converse, it will su ce to show that e b. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. Norms and metrics, normed vector spaces and metric spaces. It follows that pis an accumulation point of sif and only if each bp.
Metric spaces, topological spaces, and compactness 255 theorem a. Turns out, these three definitions are essentially equivalent. The following properties of a metric space are equivalent. Then d is a metric on r2, called the euclidean, or. Well generalize from euclidean spaces to more general spaces, such as spaces of functions. In general metric spaces, the boundedness is replaced by socalled total boundedness. Here we can think of the fr as a copy of r living inside of r2. But what is the conceptual foundation of the limit. The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as. A metric space is just a set x equipped with a function d of two variables which measures the distance between points. Paper 2, section i 4e metric and topological spaces. A metric space x is compact if every open cover of x has a. We need one more lemma before proving the classical version of ascolis theorem. A metric space is a pair x,d consisting of a set x and a metric d on x.
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