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