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. If uis an open cover of k, then there is a 0 such that for each x2kthere is a. Introduction in chapter i we looked at properties of sets, and in chapter ii we added some additional structure to a set a distance function to create a pseudomet. N is a continuous function between metric spaces m and n and that mis compact. A space x is separable if it admits a countable dense subset.
A metric space xis called totally bounded if for any 0 it can be covered by a nite number of open balls with radius. Pdf in this article, we mainly formalize in mizar 2 the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real. Characterizations of compact metric spaces france dacar, jo. On uniform continuity and compactness in pseudo metric spaces dr.
Pdf we introduce the notion of sequentially compactness on s jsmetric spaces and study the properties of sequentially compact s jsmetric spaces. We will establish various properties of compact metric spaces and provide various equivalent characterizations. Let x be a topological space and let y,d be a metric space. These proofs are merely a rephrasing of this in rudin but perhaps the di. Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces.
Jan 02, 2017 a video explaining the idea of compactness in r with an example of a compact set and a noncompact set in r. We here show how one can deduce both the kolmogorov compactness theorem and the arzelaascoli theorem from one common lemma on compactness in metric spaces. Click download or read online button to get metric space book now. The answer is yes, and the theory is called the theory of metric spaces. We prove that majority of completeness and compactness properties studied in this paper coincide for metric spaces theorem 4. Compactness in metric spacescompact sets in banach spaces and hilbert spaceshistory and motivationweak convergencefrom local to globaldirect methods in calculus of variationssequential compactnessapplications in metric spaces equivalence of compactness theorem in metric space, a subset kis compact if and only if it is sequentially compact. Definition x, d is bounded if m 0 such that d x, y. As noted, we would also like to demonstrate that the cantor set is compact, which means that any open cover for t. There are many ways to make new metric spaces from old. Handout on compactness criteria we have seen two ways to think about compactness in metric spaces. But rst, we prove that a sequentially compact space is totally bounded.
Sep 29, 2010 having a complete metric space isnt enough. The following properties of a metric space are equivalent. In general metric spaces, the boundedness is replaced by socalled total boundedness. The rst of these will be called the \ nite intersection property fip for closed sets, and turns. 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.
Chapter i normed vector spaces, banach spaces and metric spaces 1 normed vector spaces and banach spaces in the following let xbe a linear space vector space over the eld f 2fr. One of its key words is niteness, so it is closely related to compactness. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. In particular, for banach spaces, pre compactness is equivalent to being totally bounded. Defn a set k in a metric space x,d is said to be compact if each open cover of. This site is like a library, use search box in the widget to get ebook that you want. A each sequence xk in x has a convergent subsequence. Another example is the definition of distributions, which uses the space of smooth functions that are zero outside of some unspecified compact space. In all the proofs i have found, they used the bolzanoweierstrass theor. In this section we relate compactness to completeness through the idea of total boundedness in theorem 45.
Comparison of sequential compactness and limit point compactness. Metric spaces, balls, open sets, unions hints are at the end of the project. A closed bounded interval i a,b in r is totally bounded and complete, thus compact. Cauchy sequence in x has a convergent subsequence, so, by lemma 6 below. A metric space is called totally bounded if it admits a. Compactness and sequential compactness in metric spaces. Metricandtopologicalspaces university of cambridge. Xis compact and r0, then one can cover kby a nite collection of. 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.
A of open sets is called an open cover of x if every x. For example, it is easily seen that a is equivalent to the. For example r is separable q is countable, and it is dense since every real number is a limit of. Metric space download ebook pdf, epub, tuebl, mobi. Metric spaces, topological spaces, and compactness proposition a. The pseudo metric spaces which have the property that all continuous real valued functions are uniformly continuous have been studied. While compact may infer small size, this is not true in general. A subset of an open cover whose union also contains the set a is called a subcover of the original cover. Any second countable hausdor space xthat is locally compact is paracompact. For 1 i have r with usual metric and infinite metric space. A continuous function on a compact metric space is uniformly continuous. Also dont have enough examples of connected metric space. Say a metric space xis sequentially compact if every sequence in xhas a subsequence that converges in x.
There is a nice relationship between total boundedness and compactness. Compactness in metric spaces the closed intervals a. Suppose that x is a sequentially compact metric space. A continuous function of a compact metric space into a metric space is uniformly continuous. Kolmogorov compactness theorem, or frechetkolmogorov compactness theorem. This can be found in any introductory topology book, but we will not need it.
A metric space is complete if every cauchy sequence converges. In this chapter we first define the notions of a compact metric space and of a relatively compact subset of a metric space. Pdf we introduce the notion of sequentially compactness on s js metric spaces and study the properties of sequentially compact s js metric spaces. Compactness defn a collection of open sets is said to be an open cover for a set a if the union of the collection contains a. Metric spaces, topological spaces, and compactness a metric space is a set x. We do not develop their theory in detail, and we leave the veri. We say a metric space xis compact provided the following property holds. The book only gives proofs of the main properties of compact spaces using the sequence characterization of compactness, so here i reprove these main properties using the open cover point of view. Proofs of this theorem are frequently based on the arzelaascoli theorem. For the proof that i is totally bounded note that we can cover i with n. Compactness and metric spaces saul glasman october 10, 2016 first midterm will be available next friday and due the following friday.
As each pseudo metric space is normal by urysohns lemma there is f. It is proved that the following three conditions on pseudo metric space x are equivalent a every continuous real valued function on x is uniformly continuous. In this article, we mainly formalize in mizar 2 the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. It will cover the intersection of the lectures with chapter 2 of munkres.
Compactness in these notes we will assume all sets are in a metric space x. It is well known that a metric space is compact if and only if it is complete and totally bounded see, e. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. The arzel aascoli theorem gives a necessary and su cient condition for a set in c0 to be totally bounded and hence precompact. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. I need some basic examples so that i can understand easily. If x,d is a metric space and a is a nonempty subset of x, we can make a metric d a on a by putting. 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. Compactness and metric spaces math user home pages. Later we will use it to show that being sequentially compact implies compactness. Then we state and prove hausdorffs theorem of the characterization of the relatively compact subsets of a complete metric space in terms of finite and relatively compact.
A metric space is called sequentially compact if every sequence in x has a convergent subsequence. Turns out, these three definitions are essentially equivalent. If uis an open cover of k, then there is a 0 such that for each. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces. A subset, k, of m is said to be compact if and only if every open cover of k by open sets in m. The particular distance function must satisfy the following conditions. Theorem each compact set k in a metric space is closed and bounded. Y is continuous and xis compact then fx is compact. Completeness and completion compactness in metric spaces.
A metric space is just a set x equipped with a function d of two variables which measures the distance between points. A metric space x is compact if every open cover of x has a finite subcover. For 2 i have set of rational numbers with usual metric. Then the closed unit ball is closed and bounded, but not compact. Suppose kis a subset of a metric space xand k is sequentially compact. To prove this claim, we need only notice that the cantor set is bounded by 0 and 1, and recall that the closed and bounded subsets of r are exactly the compact subsets of r. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. An open cover of a metric space x is a collection countable or uncountable of open sets fu. Completeness and compactness properties in metric spaces. This volume provides a complete introduction to metric space theory for undergraduates. It is not hard to check that d is a metric on x, usually referred to as the discrete metric.
A cover is called finite if it has finitely many members. We then have the following fundamental theorem characterizing compact metric spaces. If the subset f of cx,y is totally bounded under the uniform metric corresponding to d, then f is equicontinuous under d. We wish to present two more ways to think about compactness. We then looked at some of the most basic definitions and properties of pseudometric spaces. First we recall the open cover characterization of compactness. Then by compactness of e, this open cover has a finite subcover say bx.
A metric space is a set x where we have a notion of distance. Notes on compactness these are some notes which supplement the material on compactness in the book. Compact sets in metric spaces uc davis mathematics. Remember that an open cover of kis a collection of open sets whose union covers the set k. A metric space x,d consists of a set x together with a metric d on x. U a metric space x is compact if every open cover of x has a. The third of these properties is called the triangle inequality.
Continuity convergence distance metric space theory metric spaces open sets calculus compactness minimum. For each y t there is an x t such that fx t y t there may be more than one such x t if fis not onetoone. Since fis continuous, for each p2mthere exists p 0 which may depend on p such that d mq. A uniform space is compact if and only if it is both totally bounded. Hausdorffs theorem of the characterization of the relatively compact subsets of a complete metric space in terms of finite and relatively. Since we are interested in compactness results for subsets of banach spaces, we may, and shall, concentrate our attention on total. A metric space is sequentially compact if every sequence has a convergent subsequence. Now, if you have a complete metric space where bounded sets are totally bounded, then closed and bounded subsets are compact.
A video explaining the idea of compactness in r with an example of a compact set and a noncompact set in r. Proposition each closed subset of a compact set is also. We need one more lemma before proving the classical version of ascolis theorem. On exams and in the homework, you can use whichever point of view you nd simpler to work with. Im trying to prove that every metric space is compact if and only if the space is sequentially compact.
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