Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. Find materials for this course in the pages linked along the left. Among the undergraduates here, real analysis was viewed as being one of the most dif. This theorem is the basis of many existence results in real analysis. Math 431 real analysis i solutions to homework due october 1. In general topology, a compact set mathkmath is a set for which every open cover of mathkmath contains a finite subcover of mathkmath. A space is locally compact if it is locally compact at each point. A sequence of functions f n x with domain d converges uniformly to a function fx if given any 0 there is a positive integer n such that f n x fx 0 such that q sn k1 q. A set is closed if it contains the limit of any convergent sequence within it. Definition 58 a set in rn is called compact if it is closed and bounded. We say that a is compact if for every open cover u. In mathematics, the support of a realvalued function f is the subset of the domain containing those elements which are not mapped to zero. These proofs are merely a rephrasing of this in rudin but perhaps the di. While compact may infer small size, this is not true in general.
Adoes belong to a, then we also denote it by maxaand refer to it as the maximum of a. Theorem 3 suppose that s and t are sets and that t. The most familiar is the real numbers with the usual absolute value. Note that every compact space is locally compact, since the whole space xsatis es the necessary condition. If xis locally compact and hausdor, then all compact sets in xare closed and hence if nis a compact neighborhood of xthen ncontains the closure the open intn around x. Ramanujan institute csir net jrf mathematics 17,681 views. These are some notes on introductory real analysis.
In metric spaces, a compact space glorified set math. An outer measure or exterior measure on x is a function. We emphasize that although the compact sets in r are exactly the closed and. The course assumes that the student has seen the basics of real variable theory and point set topology.
Math 431 real analysis i solutions to homework due october 1 in class, we learned of the concept of an open cover of a set s. D ris compact if and only if for any given open covering of d we can subtract a. Compactness in these notes we will assume all sets are in a metric space x. An open covering of x is a collection of open sets whose union is x. Conversely, we want to show that if s is closed and bounded, then s is compact. Real analysis, spring 2010, harvey mudd college, professor francis su. For example, once we show that a set is bounded from above, we can assert the existence of.
They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, di erentiability, sequences and series of functions, and riemann integration. If fn is a sequence of nonempty closed subsets of a sequentially compact set k such that fn. In plain english, an open cover of is a collection of open sets that cover the set. Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. Problems and solutions in real analysis series on number. R1is a set such that every open cover of e contains a finite subcover.
Since s is closed and bounded, our lemma tells us that. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. Hence, in such cases every point x2xlies in an open whose closure is compact. Then a n is a cauchy sequence and by assumption the sequence a n has a convergent subsequence. The space xis locally compact if each x2xadmits a compact neighborhood n. Assignments real analysis mathematics mit opencourseware. A of open sets is called an open cover of x if every x. An open cover of ein xis a collection of sets s whose union covers e. Each chapter has an introduction, in which some fundamental definitions and propositions are. Unlike topological concepts such as open, closed and compact sets which in principal require very little structure on the space, convexity makes use of a. Nonempty subset of the real line has intermediate point property iff it is a point, an interval, a ray, or the whole real line.
This unique book provides a collection of more than 200 mathematical problems and their detailed solutions, which contain very useful tips and skills in real analysis. Every nonempty set of real numbers that is bounded from above has a supremum, and every nonempty set of real numbers that is bounded from below has an in. Every bounded sequence of real numbers has a convergent subsequence. It is the space of continuous functions that vanish at in nity. Mod10 lec12 compact sets and its properties youtube. A continuous function maps compact sets into compact sets. We then discuss the real numbers from both the axiomatic. Compact and perfect sets we have already seen that all open sets in the real line can be written as the countable union of disjoint open intervals. Introduction to real analysis fall 2014 lecture notes.
A a collection of open subsets of r a an arbitrary set of indices. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Any discrete compact space with more than one element is disconnected. Definition of compact set if any set has a open cover and containing finite subcover than it is compact definition. The most important type of closed sets in the real line are called compact sets. As each a n is closed it follows that a2\1 k1 a n and from diam a n. Compact sets in metric spaces notes for math 703 3 such that each a n cant be nitely covered by c. We used this concept to define a compact set s as in which every infinite cover of s has a finite subcover. Compact pdf specification compact pdf is a new format that can give an additional compression of 30 to 60% on many classes of pdf beyond what is possible in pdf 1. Compactness in metric spaces the closed intervals a,b of the real line, and more generally the closed bounded subsets of rn, have some remarkable properties, which i believe you have studied in your course in real analysis. To this end, assume that fis an open cover for s n k1 a k.
The entire set of real numbers r is obviously open, and the empty set. Trench pdf 583p this is a text for a twoterm course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. This concept is used very widely in mathematical analysis. They dont include multivariable calculus or contain any problem sets. The following characterization of compact sets is fundamental compared to the sequential definition as it depends only on the underlying topology open sets 2. Math 431 real analysis i solutions to test 1 question 1. Below, you are given an open set sand a point x 2s. It is also sometimes called a distance function or simply a distance often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used we already know a few examples of metric spaces. These express functions from some set to itself, that is, with one input and one output. There is a branch of real analysis which plays a relatively modest role in pure mathematics, but is an enormously powerful device in economics. A continuous realvalued function defined on an interval in r has the intermediate.
This is a lecture notes on distributions without locally convex spaces, very basic functional analysis, lp spaces, sobolev spaces, bounded operators, spectral theory for compact self adjoint operators and the fourier transform. If for a set a, if it is a subset of union of sets then we say cover a. Dec2016 connected, compact and path connected by prof. A set s of real numbers is called compact if every sequence in s has a subsequence that converges to an element again contained in s. For example, an infinite set with discrete metric is closed and bounded, but not compact. Real analysiscompact sets wikibooks, open books for an.
The metric space x is said to be compact if every open covering has a. An open cover of a set is a collection of sets such that. They cover the properties of the real numbers, sequences and series of real numbers, limits. An alternative definition of dense set in the case of metric spaces is the following. In other words a set is compact if and only if every open cover has a finite subcover. A subset of real line is said to have intermediate point property if. Real analysis metric spaces wikibooks, open books for an. The course assumes that the student has seen the basics of real variable theory and. I have taught the beginning graduate course in real variables and functional analysis three times in the last.
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