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. A metric space is a set xtogether with a metric don it, and we will use the notation x. What topological spaces can do that metric spaces cannot82 12. Professor copsons book, which is based on lectures given to thirdyear undergraduates at the university of st andrews, provides a more leisurely treatment of metric. A metric space is a pair x, d, where x is a set and d is a metric on x. Of course, this example includes the previous one as a special case. Further, a metric space is compact if and only if each realvalued continuous function on it is bounded and attains its least and greatest values. 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 length space is called a complete length space if for all x. If v,k k is a normed vector space, then the condition du,v ku. Turns out, these three definitions are essentially equivalent. We will show in the later sections that this is actually a complete metric space and that it \contains the original metric space e. X a, there is a sequence x n in a which converges to x.
Then we call k k a norm and say that v,k k is a normed vector space. The boundary points are compact metric spaces x,d with. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. Suppose x is euclidean or hyperbolic space, then xis a complete length space.
In mathematics, a metric space is a set together with a metric on the set. The limit of a sequence in a metric space is unique. 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 metric space is just a set x equipped with a function d of two variables which measures the distance between points. 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.
Since every continuous function on a closed and bounded interval is bounded, therefore we have i i i i. A metric space y is clocally linearly connected if there exits c. Suppose x n is a convergent sequence which converges to two di. Time and space are two separate spaces and the movement in one space. Any normed vector space can be made into a metric space in a natural way. Remarks on g metric spaces and fixed point theorems fixed. Let a be a dense subset of x and let f be a uniformly continuous from a into y. Real analysismetric spaces wikibooks, open books for an. A metric space consists of a set xtogether with a function d. A point p is a limit point of the set e if every neighbourhood of p contains a point q. X r, we say that the pair m x,d is a metric space if and only if d satis.
If all neighbourhood of x contains infinitely many elements from a then x is called an accumulation point of a. A function is a metric if it satisfies the following three properties for any choice of elements. In the exercises you will see that the case m 3 proves the triangle inequality for the spherical metric of example 1. Lots of information about metric spaces can be found in marsdens.
Note that every metric space is a rectangular metric space g. In addition, the closed line segment with end points x and y consists of all points as above, but with 0 t 1. Thanks for contributing an answer to mathematics stack exchange. The length space obtained from a sequence of disjoint segments a i. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Often, if the metric dis clear from context, we will simply denote the metric space x.
Metric space topology, as the generalization to abstract spaces of the theory of sets of points on a line or in a plane, unifies many branches of classical analysis and is necessary introduction to functional analysis. In a compact metric space m let s be a set of continuous curves with the following properties. A topological space whose topology can be described by a metric is called metrizable an important source of metrics in. In particular, it is shown that for x a bounded metric space, the supremum of the mapping radii of x in. It su ces to show that c b if xis a complete metric space.
Neither of these length spaces is a geodesic space. On some fixed point results in bmetric, rectangular and b. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. Note that iff if then so thus on the other hand, let. But this follows from the corollary in the preceding section when u x. Given a set, we say is a metric space if it comes equipped with a special function that can compute the distance between any two points of. Each compact metric space is complete, but the converse is false. Diameter of metric spaces mathematics stack exchange. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A metric space is termed a lengthmetric space if the distance between any two points in it equals the infimum of the lengths of all the paths joining them. Chapter 1 metric spaces islamic university of gaza. Analysis on metric spaces summer school, lake arrowhead.
If the space has the stronger property that there always exists a path that achieves the infimum of length a geodesic then it may be called a geodesic metric space or geodesic space. A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. A topological space whose topology can be described by a metric is called metrizable. We just saw that the metric space k 1 isometrically embeds into 2 k in fact, a stronger result can be shown. However the converse is not necessarily true 11, examples 2. To qualify as a metric, the function dmust satisfy certain properties. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. In verifying that this really is a metric space, the only di. Compared to the shortest ruler, how much longer is the longest ruler. The most familiar is the real numbers with the usual absolute value. Add justi cations to the following proof of this fact. A metric induces a topology on a set, but not all topologies can be generated by a metric. 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. 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 in which the points are functions are used to solve di erential equations and other topics in analysis. Tools are developed for estimating the mapping radius of a metric space x in a metric space y. The following properties of a metric space are equivalent. Other metric spaces are the taxicab space and the torus. Aug 26, 2012 given a set, we say is a metric space if it comes equipped with a special function that can compute the distance between any two points of.
Theorem in a any metric space arbitrary intersections and finite unions of closed sets are closed. Ifi, j, and kare chosen to be orthonormal, the resulting metric is the standard metric on s3 i. Here d is the metric on x, that is, dx, y is regarded as the distance from x to y. U nofthem, the cartesian product of u with itself n times. If xis a complete metric space with property c, then xis compact. One of these balls contains in nitely many points of s, and so does its closure, say x1 b12y1. Metricandtopologicalspaces university of cambridge. In other words, no sequence may converge to two di. Gromovhausdorff space each point is a compact metric space. 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. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. But avoid asking for help, clarification, or responding to other answers. Completion of a metric space a metric space need not be complete.
185 494 1157 132 222 912 344 1042 448 1222 876 809 886 379 503 1612 866 1205 250 1 1475 1571 25 1046 687 1028 1291 664 1014 962 1002 697 280 1265 1082 666 950 1334 889 1290