Introduction when we consider properties of a reasonable function, probably the. In chapter 2, we defined fuzzy soft metric space with suitable illustrations. What topological spaces can do that metric spaces cannot. A metric space is a set x where we have a notion of distance.
A metric space is a set xtogether with a metric don it, and we will use the notation x. Exercises for mat2400 metric spaces mathematicians like to make general theories. We do not develop their theory in detail, and we leave the veri. Then d is a metric on r2, called the euclidean, or. 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. A sequence x n in x is called a cauchy sequence if for any. In chapter 1, the basic definitions, examples, properties and theorems are given which are used for throughout the dissertation. Real analysismetric spaces wikibooks, open books for an. The limit of a sequence in a metric space is unique. We also have the following simple lemma lemma 3 a subset u of a metric space is open if and only if it is a neighbor hood of each of its points. A set, whose elements we shall call points, is said to be a metric space if with any two points and of there is associated a real number, called the distance from to.
In your first calculus class, you encountered the notion of a limit of a function at a point. A subset k of x is compact if every open cover of k has a. Variety of examples along with real life applications have been provided to understand and appreciate the beauty of metric spaces. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. 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. A metric space is complete if every cauchy sequence has a limit. If is the real line with usual metric, then remarks. A pathconnected space is a stronger notion of connectedness, requiring the structure of a path. Notes on metric spaces these notes are an alternative to the textbook, from and including closed sets and open sets page 58 to and excluding cantor sets page 95 1 the topology of metric spaces assume m is a metric space with distance function d. A metric space is called complete if every cauchy sequence converges to a limit. X r which measures the distance dx,y beween points x,y.
The metric is often regarded as a distance function. The following standard lemma is often useful, and makes explicit a bit of intuition. 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. Metric spaces math 441, summer 2009 we begin this class by a motivational introduction to metric spaces. Then this does define a metric, in which no distinct pair of points are close. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric.
In other words, no sequence may converge to two di. X y between metric spaces is continuous if and only if f. First, suppose f is continuous and let u be open in y. Show from rst principles that if v is a vector space over r or c then for any set xthe space 5. 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 survey article pdf available in middle east journal of scientific research 1112 february 2011 with 2,068 reads how we measure reads. For the theory to work, we need the function d to have properties similar. The fact that every pair is spread out is why this metric is called discrete. Ais a family of sets in cindexed by some index set a,then a o c. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. An open neighbourhood of a point p is the set of all points within of it. A pathcomponent of x is an equivalence class of x under the equivalence relation which makes x equivalent to y if there is a path from x to y.
Often, if the metric dis clear from context, we will simply denote the metric space x. Vg is a linear space over the same eld, with pointwise operations. Metric spaces notes these are updated version of previous notes. Notes of metric spaces these notes are related to section iv of b course of mathematics, paper b. The halfopen interval 0,1 is a neighbourhood of every point in it except for 0. The particular distance function must satisfy the following conditions.
The function dis called the metric, it is also called the distance function. Metric spaces generalize and clarify the notion of distance in the real line. Informally, 3 and 4 say, respectively, that cis closed under. Turns out, these three definitions are essentially equivalent. U nofthem, the cartesian product of u with itself n times. In calculus on r, a fundamental role is played by those subsets of r which are intervals. These will be the standard examples of metric spaces. Suppose x n is a convergent sequence which converges to two di. The most familiar is the real numbers with the usual absolute value. Metric spaces, open balls, and limit points definition. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. A metric space is compact if and only if it is complete and totally bounded.
Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. Regrettably mathematical and statistical content in pdf files is unlikely to be. The following properties of a metric space are equivalent. Metric space download ebook pdf, epub, tuebl, mobi. Every closed subset of a compact space is itself compact. Examples of compact metric spaces include the closed interval 0,1 with the absolute value metric, all metric spaces with finitely many points, and the cantor set. It saves work to do things once and for all, but more importantly, often it makes things clearer.
Metricandtopologicalspaces university of cambridge. In particular, whenever we talk about the metric spaces rn without explicitly specifying the metrics, these are the ones we are talking about. You have met or you will meet the concept of a normed vector space both in algebra and analysis courses. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Definition a metric space is a set x together with a function d called a metric or distance function which assigns a real number d. Moreover the concepts of metric subspace, metric superspace. This volume provides a complete introduction to metric space theory for undergraduates. It turns the spotlight on the salient points of the theory, and shows what is the important questions. We then have the following fundamental theorem characterizing compact metric spaces. Let us take a look at some examples of metric spaces. Some important properties of this idea are abstracted into. The observation above that the given metric on rn gives the usual notion of distance is what. 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 distance as far as possible, illustrating the text with examples and naturally arising questions.
The analogues of open intervals in general metric spaces are the following. The concepts of metric and metric space are generalizations of the idea of distance in euclidean space. In topology, a secondcountable space also called a completely separable space is a topological space having a countable base it is well known that a secondcountable space is separable. 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. Will have many more examples later, as the course proceeds. A path from a point x to a point y in a topological space x is a continuous function. Cauchy sequences and complete metric spaces lets rst consider two examples of convergent sequences in r. These notes are collected, composed and corrected by atiq ur rehman, phd. Well give some examples and define continuity on metric spaces, then show how continuity can be stated without reference to metrics.
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