Compact metric spaces pdf

A metric space is called sequentially compact if every sequence in x has a convergent subsequence. I dont like the wikipedia quote, as it sort of suggests that sequences in compact spaces must be convergent. Suppose kis a subset of a metric space xand k is sequentially compact. Say a metric space xis sequentially compact if every sequence in xhas a subsequence that converges in x. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. A metric space is compact iff every sequence has a convergent subsequence. A metric space which is totally bounded and complete is also sequentially compact. A metric space is sequentially compact if every sequence has a convergent subsequence. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. A metric space is complete if every cauchy sequence converges. It is stronger then usual continuity at every point because here depends only on the and not on the point nonexample. That is, a subset k of m is compact if and only if every cover of k by open subsets of m has a.

In general metric spaces, the boundedness is replaced by socalled total boundedness. Every topological space x is an open dense subspace of a compact space having at most one point more than x, by the alexandroff onepoint compactification. Every closed subset of a compact metric space is compact. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex. Jan 02, 2017 a video explaining the idea of compactness in r with an example of a compact set and a non compact set in r. In the subsequent sections we discuss the proof of the lemmata. A metric space is a set x where we have a notion of distance. We do not develop their theory in detail, and we leave the veri. Let d denote either the square metric or the euclidean metric on rn. X of compact metric spaces, they satisfy the eilenbergsteenrod axioms 7, chpt. Using it show that the product metric space of two compact metric spaces is compact where the product of two metric spaces. A metric space x is compact if every open cover of x has a finite subcover. A subset, k, of m is said to be compact if and only if every open cover of k by open sets in m has a finite subcover.

A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. Cauchy sequence in x has a convergent subsequence, so, by lemma 6 below. Suppose that x is a sequentially compact metric space. Computable operations on compact subsets of metric spaces with. 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. Pdf cuniform distribution on compact metric spaces. Metricandtopologicalspaces university of cambridge. If xis a compact metric space, it has a countable dense subset. We then have the following fundamental theorem characterizing compact metric spaces.

The following properties of a metric space are equivalent. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. Pdf we introduce the notion of sequentially compactness on s jsmetric spaces and study the properties of sequentially compact s jsmetric 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 metric space is sequentially compact if and only if every in. Characterizations of compactness for metric spaces 3 the proof of the main theorem is contained in a sequence of lemmata which we now state.

Here is another useful property of compact metric spaces, which will eventually be generalized even further, in e below. The equivalence between closed and boundedness and compactness is valid in nite dimensional euclidean. My interest in developing the theory of compact quantum metric spaces was. The product of any collection of compact spaces is compact. 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. If x is a lipschitz manifold or m chains and forms 5. This is true if, for instance, we consider the space of compact subsets of a complete and perfect metric space see 27, the paragraph below lemma 1. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. Both are functorial on the category of compact metric spaces and lipschitz maps, and when extended to the pairs a. There are many ways to make new metric spaces from old. It is also true with respect to the gromovhausdor metric 22, corollary 5. Schep in this note we shall present a proof that in a metric space x. Definition a metric space x, d is said to be complete if every cauchy sequence in x converges to some point of x. Compact sets in metric spaces uc davis mathematics.

While compact may infer small size, this is not true in general. A subset of a metric space m is said to be compact if it is a compact subset of the topological space m. 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. Every continuous function on a compact set is uniformly continuous. Informally, 3 and 4 say, respectively, that cis closed under. This is tychonoffs theorem, which is equivalent to the axiom of choice.

Completions a notcomplete metric space presents the di culty that cauchy sequences may fail to converge. Reasonably, we want to repair this situation, and in as economical way as possible. The particular distance function must satisfy the following conditions. A metric space which is sequentially compact is totally bounded and complete. 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. Characterizations of compact metric spaces france dacar, jo. A subspace f of cx,rn is compact if and only if it is closed, bounded under the sup metric. How to understand the concept of compact space mathoverflow. Turns out, these three definitions are essentially equivalent. A metric space is compact if and only if it is sequentially compact, but this does not hold for nonmetrizable topological spaces.

Chains and forms in compact metric spaces pdf free download. 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. Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. In almost all applications, locally compact spaces are indeed also hausdorff. A subset k of x is compact if every open cover of k has a. A metric space x is compact if every open cover of x has a.

A sequentially compact subset of a metric space is bounded and closed. In the case of metric spaces, the compactness, the countable compactness and the sequential compactness are equivalent. Ais a family of sets in cindexed by some index set a,then a o c. We give a brief survey of many of the highlights of our present understanding of the young subject of quantum metric spaces, and of quantum gromovhausdor. Given 2 n, let sn be a nite set of points xj such that fbxjg covers x. It is not necessarily the case that the set of limit points of is the closure of. The product metric space of two compact metric spaces is. If uis an open cover of k, then there is a 0 such that for each x2kthere is a. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. This equivalence is a consequence of the facts that compact subsets of hausdorff spaces are closed, and closed subsets of compact spaces are compact.