A space x is separable if it admits a countable dense subset. The main reason for taking up such a project is to have an electronic backup of my own handwritten solutions. Later we will use it to show that being sequentially compact implies compactness. The general class of metric spaces is large, and contains many ill behaved examples one of which is any set endowed with the discrete metric good for gaining intuition, a nightmare to work with. Lecture notes assignments download course materials. Compactness applied topology applied topology is intended to expose students with majors in mathematics or the sciences to basic topology and its recent applications. 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. Suppose kis a subset of a metric space xand k is sequentially compact. A sequentially compact space is a space such that every sequence. We are going to therefore introduce two regularity conditions that give us nice 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.
A metric space is a special kind of quasi metric space. Recall that every normed vector space is a metric space, with the metric dx. A metric space is a set in which we can talk of the distance between any two of its elements. Open covers, compact sets 1 insu ciency of open and closed sets consider the question of embedding metric spaces inside of one another. A metric space is a set x where we have a notion of distance. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible. Metric and metrizable spaces gmu math 631 spring 2011. A cover oadmits a nite subcover i there is a nite set o o such that x s o2o o. Lecture 3 complete metric spaces 1 complete metric spaces 1. For all of the lecture notes, including a table of contents, download the following file pdf. For example r is separable q is countable, and it is dense since every real number is a limit of. If the subset f of cx,y is totally bounded under the uniform metric corresponding to d, then f is equicontinuous under d. A metric space xis called totally bounded if for any 0 it can be covered by a nite number of open balls with radius.
If uis an open cover of k, then there is a 0 such that for each. Pointset theory, compactness, completeness, connectedness, total boundedness, density, category, uniform continuity and convergence, stoneweierstrass theorem, fixedpoint theorems. The definition below imposes certain natural conditions on the distance between the points. A metric space is complete if every cauchy sequence converges.
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. 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. We adjust methods of computable model theory to e ective analysis. The lecture notes were taken by a student in the class. For any metric space x, d, the following are equivalent. Basic inequality on a bmetric space and its applications ncbi. Characterizations of compact metric spaces france dacar, jo. 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. A metric space is sequentially compact if every sequence has a convergent subsequence. 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. In mathematics, more specifically in general topology, compactness is a property that.
What links here related changes upload file special pages permanent link page. A onepage selfcontained complete proof that metric spaces are paracompact. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. International journal of mathematics and mathematical sciences.
A of open sets is called an open cover of x if every x. The rst of these will be called the \ nite intersection property fip for closed sets, and turns. Y is continuous and xis compact then fx is compact. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. This site is like a library, use search box in the widget to get ebook that you want. A subset s of a metric space m is bounded if there are a. Lecture notes analysis ii mathematics mit opencourseware. 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. These proofs are merely a rephrasing of this in rudin but perhaps the di. That is to say, suppose that we have a metric spaces x. In general metric spaces, the boundedness is replaced by socalled total boundedness.
Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. X, d is complete and totally bounded this is also equivalent to compactness for uniform spaces. A subset of an open cover whose union also contains the set a is called a subcover of the original cover. A metric space x, d is called complete if every cauchy sequence in x converges. As a euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. However, the product of a paracompact space and a compact space is always paracompact. There are many ways to make new metric spaces from old. A metric space is sequentially compact if and only if every in. The goal of these notes is to construct a complete metric space which contains x as a subspace and which is the \smallest space with respect to these two properties. Say a metric space xis sequentially compact if every sequence in xhas a subsequence that converges in x. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line.
A topological space x is said to be limit point compact if every infinite subset of x has a limit point in x, and countably compact if every countable open cover has a finite subcover. Math 556 metric spaces course description from bulletin. The prerequisite is linear algebra together with an introduction to proofs. Can choose a metric suited to particular purpose metrics may be complicated, while the topology may be simple can study families of metrics on a xed topological space. A solutions manual for topology by james munkres 9beach. A metric space x is compact if every open cover of x has a finite subcover. Prove that any nonempty set x is made into a complete metric space if it is given the discrete metric. A metric space x is compact if every open cover of x has a. A disconnection of a set a in a metric space x,d consists of two nonempty sets a 1, a 2 whose disjoint union is a and each is open relative to a. Compactness and sequential compactness in metric spaces. We use index sets and in nitary logic to obtain classi cationtype results for compact computable metric spaces. We need one more lemma before proving the classical version of ascolis theorem. If the topology t on x is generated by the ometric d.
The idea of closure is strongly linked to the idea of a dense subset. Tutorial sheets are available online as pdf files from the resources page on the friday of the week before the tutorial takes place. You should have received an email detailing the format along with details of some changes to the marking of continuous assessment. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. To prove the converse, it will su ce to show that e b. The typical distance used is the euclidean distance, defined as. Metric space download ebook pdf, epub, tuebl, mobi. Tychono is one of the most important results in topology and it is one of the main tools for establishing compactness in economics settings such as repeated games and dynamic macro models. Handout on compactness criteria we have seen two ways to think about compactness in metric spaces. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. Compactness in these notes we will assume all sets are in a metric space x. Xthe number dx,y gives us the distance between them. The particular distance function must satisfy the following conditions.
In this paper we consider, discuss, improve and generalize recent fixed point results for mappings in bmetric, rectangular metric and brectangular metric spaces established by dukic et al. The completion of a metric space northwestern university. If uis an open cover of k, then there is a 0 such that for each x2kthere is a. A quasi metric space x with quasi metric d will be denoted by x, d. Completely metrizable spaces are often called topologically complete. Wesaythatasequencex n n2n xisacauchy sequence ifforall0 thereexistsann.
Metric spaces, topological spaces, and compactness 255 theorem a. Metricandtopologicalspaces university of cambridge. One of its key words is niteness, so it is closely related to compactness. Click download or read online button to get metric space book now. In the subsequent sections we discuss the proof of the lemmata. The resulting space will be denoted by xand will be called the completion of xwith respect to d. X is closed and bounded as a subset of any metric space whose restricted. In mathematics, a metric space is a set together with a metric on the set. 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.
Remember that an open cover of kis a collection of open sets whose union covers the set k. A space x is called countably compact if every countable open cover of x has a. Metric spaces, balls, open sets, unions hints are at the end of the project. Compact sets in metric spaces uc davis mathematics. Complete and incomplete metric spaces part 1 youtube. Actually, the proof works for any firstcountable space that is a countably compact space, i. But rst, we prove that a sequentially compact space is totally bounded. Characterizations of compactness for metric spaces 3 the proof of the main theorem is contained in a sequence of lemmata which we now state.
Sometimes, when no ambiguity is possible, the space x, d may be denoted simply by x. I discuss tychono in somewhat more detail in the tychono notes. The following properties of a metric space are equivalent. This is an ongoing solutions manual for introduction to metric and topological spaces by wilson sutherland 1. We first prove one of the most basic inequalities on a bmetric space. Lecture notes functional analysis ws 201220 february, 20. Contents i normed vector spaces, banach spaces and metric spaces5. Suppose that the sequence did not have a convergent subsequence, that is, does not have a cluster point. Since the conclusion of the baire category theorem is purely topological, it applies to these spaces as well. Hence countably compact metric spaces are equivalently compact metric spaces. While compact may infer small size, this is not true in general.
Introduction to metric and topological spaces by wilson. A cover is called finite if it has finitely many members. We wish to present two more ways to think about compactness. 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. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. The pseudo metric spaces which have the property that all continuous real valued functions are uniformly continuous have been studied. Rn as a metric space, we have in mind the euclidean metric, unless another metric is speci. Let x be a topological space and let y,d be a metric space. 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 old fashioned terminology, the closure of ais its set of closure points3. Also recal the statement of lemma a closed subspace of a complete metric space is complete. X is compact i every open cover admits a nite subcover. Turns out, these three definitions are essentially equivalent. What links here related changes upload file special pages permanent link.
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