Find materials for this course in the pages linked along the left. An alternative definition of dense set in the case of metric spaces is the following. The space xis locally compact if each x2xadmits a compact neighborhood n. Theorem 3 suppose that s and t are sets and that t. 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. 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. The most important type of closed sets in the real line are called compact sets. They cover the properties of the real numbers, sequences and series of real numbers, limits. Below, you are given an open set sand a point x 2s. 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. D ris compact if and only if for any given open covering of d we can subtract a. A continuous function maps compact sets into compact sets. Real analysis metric spaces wikibooks, open books for an.
The course assumes that the student has seen the basics of real variable theory and point set topology. An open cover of a set is a collection of sets such that. Conversely, we want to show that if s is closed and bounded, then s is compact. An open covering of x is a collection of open sets whose union is x. This concept is used very widely in mathematical analysis.
This theorem is the basis of many existence results in real analysis. For example, once we show that a set is bounded from above, we can assert the existence of. 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. Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1.
It is the space of continuous functions that vanish at in nity. Definition 58 a set in rn is called compact if it is closed and bounded. Mod10 lec12 compact sets and its properties youtube. An outer measure or exterior measure on x is a function.
Definition of compact set if any set has a open cover and containing finite subcover than it is compact definition. We then discuss the real numbers from both the axiomatic. 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. Every bounded sequence of real numbers has a convergent subsequence.
They dont include multivariable calculus or contain any problem sets. These are some notes on introductory real analysis. Problems and solutions in real analysis series on number. Then a n is a cauchy sequence and by assumption the sequence a n has a convergent subsequence. Hence, in such cases every point x2xlies in an open whose closure is compact. We used this concept to define a compact set s as in which every infinite cover of s has a finite subcover. The following characterization of compact sets is fundamental compared to the sequential definition as it depends only on the underlying topology open sets 2. In plain english, an open cover of is a collection of open sets that cover the set. Compact sets in metric spaces notes for math 703 3 such that each a n cant be nitely covered by c. The metric space x is said to be compact if every open covering has a. 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. If for a set a, if it is a subset of union of sets then we say cover a. 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 these notes we will assume all sets are in a metric space x.
Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. Real analysiscompact sets wikibooks, open books for an. In other words a set is compact if and only if every open cover has a finite subcover. Adoes belong to a, then we also denote it by maxaand refer to it as the maximum of a. The most familiar is the real numbers with the usual absolute value. Dec2016 connected, compact and path connected by prof. These proofs are merely a rephrasing of this in rudin but perhaps the di. Among the undergraduates here, real analysis was viewed as being one of the most dif. A space is locally compact if it is locally compact at each point.
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. As each a n is closed it follows that a2\1 k1 a n and from diam a n. 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. In metric spaces, a compact space glorified set math. A continuous realvalued function defined on an interval in r has the intermediate. We emphasize that although the compact sets in r are exactly the closed and. An open cover of ein xis a collection of sets s whose union covers e. Nonempty subset of the real line has intermediate point property iff it is a point, an interval, a ray, or the whole real line. Assignments real analysis mathematics mit opencourseware. While compact may infer small size, this is not true in general. Ramanujan institute csir net jrf mathematics 17,681 views. 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.
Since s is closed and bounded, our lemma tells us that. Math 431 real analysis i solutions to test 1 question 1. Any discrete compact space with more than one element is disconnected. 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. Each chapter has an introduction, in which some fundamental definitions and propositions are. To this end, assume that fis an open cover for s n k1 a k. For example, an infinite set with discrete metric is closed and bounded, but not compact. We say that a is compact if for every open cover u. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. I have taught the beginning graduate course in real variables and functional analysis three times in the last. Note that every compact space is locally compact, since the whole space xsatis es the necessary condition. Real analysis, spring 2010, harvey mudd college, professor francis su.
A a collection of open subsets of r a an arbitrary set of indices. A of open sets is called an open cover of x if every x. The course assumes that the student has seen the basics of real variable theory and. 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.
In mathematics, the support of a realvalued function f is the subset of the domain containing those elements which are not mapped to zero. There is a branch of real analysis which plays a relatively modest role in pure mathematics, but is an enormously powerful device in economics. Introduction to real analysis fall 2014 lecture notes. If fn is a sequence of nonempty closed subsets of a sequentially compact set k such that fn. A subset of real line is said to have intermediate point property if.
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. 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. These express functions from some set to itself, that is, with one input and one output. 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. A set is closed if it contains the limit of any convergent sequence within it. 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.
1310 341 1010 190 755 892 1021 288 232 1209 1481 1384 204 346 935 576 872 950 1165 1167 531 1521 72 1206 431 136 756 791 876 197 1676 1558 791 635 783 39 1064 659 579 761