X a, there is a sequence x n in a which converges to x. Xis compact, and x j2kis a cauchy sequence, then there exists x2ksuch that lim j. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric. Definition a metric space x, d is said to be complete if every cauchy sequence in x converges to some point of x. A of open sets is called an open cover of x if every x.
A subset k of a metric space x is said to be compact if every open cover of k has a finite subcover. Felix hausdorff chose the name metric space in his influential book from 1914. Turns out, these three definitions are essentially equivalent. The worst you could have done to take an infinite number of steps would have been to traverse the whole space so your balls cover the space, but then compactness would. Note that iff if then so thus on the other hand, let. Any second countable hausdor space xthat is locally compact is paracompact. Pdf in this paper we recall the definition of standard fuzzy metric space then we define a compact standard fuzzy metric space and ftotally. A metric space x is compact if every open cover of x has a. Working intuitively for now see below for the precise version, lets distinguish two similar but distinct ideas.
For any metric space x, d the following conditions are equivalent. Compact metric space yongheng zhang when we say a metric space xis compact, usually we mean any open covering of xhas a nite subcovering. Since the set of the centres of these balls is finite, it has finite diameter, from. Let a be a dense subset of x and let f be a uniformly continuous from a into y. How to understand the concept of compact space mathoverflow. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. M and r0, the open ball with center x of radius r is the set bx,r. Then d is a metric on r2, called the euclidean, or.
M dx,y 0 there exist finitely many open balls of radius r whose union covers m. Xis compact, and x j2kis a sequence then a subsequence of the x j is cauchy. I was recently presented this problem from a course on topology half of which i could work out but the other half is a mystery. A compact space decomposition for effective metric. For any metric space x, d, the following are equivalent. Mat 314 lecture notes 3 of course, for 0 pdf available in proceedings of the american mathematical society 854. If uis an open cover of k, then there is a 0 such that for each.
The last metric space notion that we will use is the notion of compact sets. This can be found in any introductory topology book, but we will not need it. Compact sets in metric spaces uc davis mathematics. Suppose kis a subset of a metric space xand k is sequentially compact. Often, if the metric dis clear from context, we will simply denote the metric space x. Cauchy sequence in x has a convergent subsequence, so, by lemma 6 below. The slightly odd definition of a compact metric space is as follows. Definition 23 k c m is compact if, for every open covering f of k there exists a finite. Pdf compact nonlinear mappings in metric spaces researchgate. Also any subset of a metric space is a metric space.
A metric space is complete if every cauchy sequence converges. In the case of metric spaces, the compactness, the countable compactness and the sequential compactness are equivalent. A metric space is sequentially compact if every sequence has a convergent subsequence. Every open subset of a complete and perfect space contains a cantor space. Baire category theorem is proved and, as an application, it is shown that continuous, nowhere di erentiable functions form a set of second category in the space of continuous functions. Xthe number dx,y gives us the distance between them. 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.
X we say that a is totally bounded if, for any 0, a can be covered nite number of balls of radius. A metric space is a set xtogether with a metric don it, and we will use the notation x. Suppose that x is a sequentially compact metric space. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Separable complete metric spaces occurring in mathematical practice are usually computable. We would like to show you a description here but the site wont allow us. Honors advanced calculus and linear algebra metric. We just saw that the metric space k 1 isometrically embeds into 2 k in fact, a stronger result can be shown. 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. The equivalence between closed and boundedness and compactness is valid in nite dimensional euclidean. As a euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets.
Characterizations of compact metric spaces france dacar, jo. Characterizations of compactness for metric spaces 3 the proof of the main theorem is contained in a sequence of lemmata which we now state. A metric space is a cantor space if it is compact, perfect and totally disconnected. A metric space is called sequentially compact if every sequence in x has a convergent subsequence. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. Recall that every compact metric space is separable and complete. In the subsequent sections we discuss the proof of the lemmata. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed nball. The following properties of a metric space are equivalent.
Norms and metrics, normed vector spaces and metric spaces. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. Say a metric space xis sequentially compact if every sequence in xhas a subsequence that converges in x. Request pdf a compact space decomposition for effective metric indexing the metric space model abstracts many proximity search problems, from nearestneighbor classifiers to textual and. A onepage selfcontained complete proof that metric spaces are paracompact. A compact subspace c of a metric space m is bounded. Finally in section 5 we study complete metric spaces. A metric space is sequentially compact if and only if every in. 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 spaces than. Ccould be replaced here by any subset of v that is not a vector subspace of v i. In general metric spaces, the boundedness is replaced by socalled total boundedness. Pdf concepts of compact linear mappings and measure of non compactness for. An open cover for a is a collection o of open sets whose union contains a.
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