If you are trying seriously to learn the subject, give them a look when you have the chance. 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. This is an isometry when r is given the usual metric and r2 is given the 2dimensional euclidean metric, but not. Vg is a linear space over the same eld, with pointwise operations. The metric is often regarded as a distance function. Let aand bbe irrational numbers such that a metric spaces of functions. Let us take a look at some examples of metric spaces. We then have the following fundamental theorem characterizing compact metric spaces. Since every continuous function on a closed and bounded interval is bounded, therefore we have i i i i. Chapter 9 the topology of metric spaces uci mathematics. X y between metric spaces is continuous if and only if f. In other words, no sequence may converge to two di.
A metric space x is compact if every open cover of x has a. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. Exercises for mat2400 metric spaces mathematicians like to make general theories. Also included are several worked examples and exercises.
Properties of open subsets and a bit of set theory16 3. The analogues of open intervals in general metric spaces are the following. The following properties of a metric space are equivalent. This, instead of 8xx2rx2 0 one would write just 8xx2 0. Sometimes restrictions are indicated by use of special letters for the variables. It is also sometimes called a distance function or simply a distance. Consider q as a metric space with the usual metric. The most familiar is the real numbers with the usual absolute value. Examples of metrics, elementary properties and new metrics from old ones problem 1.
In calculus on r, a fundamental role is played by those subsets of r which are intervals. Lecture 3 complete metric spaces 1 complete metric spaces 1. 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. Definition and fundamental properties of a metric space. Ais a family of sets in cindexed by some index set a,then a o c. 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. Let aand bbe irrational numbers such that a 0, the open ball of radius. It saves work to do things once and for all, but more importantly, often it makes things clearer. A metric space is a pair x, d, where x is a set and d is a. A pair, where is a metric on is called a metric space. 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. Some important properties of this idea are abstracted into. A metric space is a pair x, d, where x is a set and d is a metric on x.
A metric space is compact if and only if it is complete and totally bounded. The fact that every pair is spread out is why this metric is called discrete. Then this does define a metric, in which no distinct pair of points are close. Bacit is because there are at least two points for which in most situations this doesnt happen. It turns the spotlight on the salient points of the theory, and shows what is the important questions. Mathematical proof or they may be 2place predicate symbols.
As a formal logical statements, this theorem can be written in the following form. Uniform metric let be any set and let define particular cases. If is the real line with usual metric, then remarks. Informally, 3 and 4 say, respectively, that cis closed under. Show that the manhatten metric or the taxicab metric. A subset k of x is compact if every open cover of k has a. Moreover the concepts of metric subspace, metric superspace. Here we can think of the fr as a copy of r living inside of r2. For the theory to work, we need the function d to have properties similar to the distance. A metric space is complete if every cauchy sequence has a limit. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible. These observations lead to the notion of completion of a metric. Metric space topology spring 2016 selected homework. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur.
In particular, whenever we talk about the metric spaces rn without explicitly specifying the metrics, these are the ones we are talking about. A metric space is a set x where we have a notion of distance. Some students, especially mathematically inclined ones, love these books, but others nd them hard to read. Turns out, these three definitions are essentially equivalent. Introduction when we consider properties of a reasonable function, probably the. Wesaythatasequencex n n2n xisacauchy sequence ifforall0 thereexistsann. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Variety of examples along with real life applications have been provided to understand and appreciate the beauty of metric spaces.
Free and bound variables 3 make this explicit in each formula. Cauchy sequences and complete metric spaces lets rst consider two examples of convergent sequences in r. We do not develop their theory in detail, and we leave the veri. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric.
Every closed subset of a compact space is itself compact. For the theory to work, we need the function d to have properties similar. Thus we could have solved the example of approximating ft t by using an or. Suppose x n is a convergent sequence which converges to two di. Thetriangularinequalityis awellknownresultfromlinearalgebra,knownasthecauchyschwartzinequality. The set of rational numbers q is a dense subset of r. Metricandtopologicalspaces university of cambridge. A particular case of the previous result, the case r 0, is that in every metric space singleton sets are closed. A metric space is called complete if every cauchy sequence converges to a limit. The limit of a sequence in a metric space is unique. Metric spaces and some basic topology ii 1x 1y d x.
The observation above that the given metric on rn gives the usual notion of distance is what. I have included 295 completely worked out examples to illustrate and clarify all. This video discusses an example of particular metric space that is complete. Then f is continuous on x iff f 1o is an open set in x whenever o is an open set in y. Metrics on spaces of functions these metrics are important for many of the applications in. You may have realised from your work on exercise 1. As we said, the standard example of a metric space is rn, and r, r2, and r3 in particular. The particular distance function must satisfy the following conditions. X r which measures the distance dx,y beween points x,y. 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.
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