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 general topology, a compact set mathkmath is a set for which every open cover of mathkmath contains a finite subcover of mathkmath. Hence, in such cases every point x2xlies in an open whose closure is compact. 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.
We used this concept to define a compact set s as in which every infinite cover of s has a finite subcover. Then a n is a cauchy sequence and by assumption the sequence a n has a convergent subsequence. In plain english, an open cover of is a collection of open sets that cover the set. The metric space x is said to be compact if every open covering has a. Find materials for this course in the pages linked along the left. For example, an infinite set with discrete metric is closed and bounded, but not compact. Math 431 real analysis i solutions to test 1 question 1. D ris compact if and only if for any given open covering of d we can subtract a.
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. Ramanujan institute csir net jrf mathematics 17,681 views. Compactness in these notes we will assume all sets are in a metric space x. 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. An open covering of x is a collection of open sets whose union is x.
An open cover of ein xis a collection of sets s whose union covers e. 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. We emphasize that although the compact sets in r are exactly the closed and. While compact may infer small size, this is not true in general. Theorem 3 suppose that s and t are sets and that t. These proofs are merely a rephrasing of this in rudin but perhaps the di. These are some notes on introductory real analysis. 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. They cover the properties of the real numbers, sequences and series of real numbers, limits. It is the space of continuous functions that vanish at in nity. These express functions from some set to itself, that is, with one input and one output. R1is a set such that every open cover of e contains a finite subcover. Real analysis, spring 2010, harvey mudd college, professor francis su.
In other words a set is compact if and only if every open cover has a finite subcover. If fn is a sequence of nonempty closed subsets of a sequentially compact set k such that fn. The most familiar is the real numbers with the usual absolute value. The entire set of real numbers r is obviously open, and the empty set.
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. 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. Among the undergraduates here, real analysis was viewed as being one of the most dif. 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. The course assumes that the student has seen the basics of real variable theory and point set topology. The following characterization of compact sets is fundamental compared to the sequential definition as it depends only on the underlying topology open sets 2. An alternative definition of dense set in the case of metric spaces is the following. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Dec2016 connected, compact and path connected by prof. Definition of compact set if any set has a open cover and containing finite subcover than it is compact definition. We say that a is compact if for every open cover u. If for a set a, if it is a subset of union of sets then we say cover a. Assignments real analysis mathematics mit opencourseware.
Problems and solutions in real analysis series on number. An outer measure or exterior measure on x is a function. The course assumes that the student has seen the basics of real variable theory and. In mathematics, the support of a realvalued function f is the subset of the domain containing those elements which are not mapped to zero. This theorem is the basis of many existence results in real analysis. The most important type of closed sets in the real line are called compact sets. Below, you are given an open set sand a point x 2s. A continuous function maps compact sets into compact sets. This concept is used very widely in mathematical analysis. Adoes belong to a, then we also denote it by maxaand refer to it as the maximum of a. Introduction to real analysis fall 2014 lecture notes. A space is locally compact if it is locally compact at each point. As each a n is closed it follows that a2\1 k1 a n and from diam a n.
Any discrete compact space with more than one element is disconnected. Real analysiscompact sets wikibooks, open books for an. Real analysis metric spaces wikibooks, open books for an. A a collection of open subsets of r a an arbitrary set of indices. They dont include multivariable calculus or contain any problem sets. Mod10 lec12 compact sets and its properties youtube.
Every bounded sequence of real numbers has a convergent subsequence. An open cover of a set is a collection of sets such that. To this end, assume that fis an open cover for s n k1 a k. 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. Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. The space xis locally compact if each x2xadmits a compact neighborhood n. Compact sets in metric spaces notes for math 703 3 such that each a n cant be nitely covered by c.
A continuous realvalued function defined on an interval in r has the intermediate. For example, once we show that a set is bounded from above, we can assert the existence of. Nonempty subset of the real line has intermediate point property iff it is a point, an interval, a ray, or the whole real line. Since s is closed and bounded, our lemma tells us that. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. Conversely, we want to show that if s is closed and bounded, then s is compact. There is a branch of real analysis which plays a relatively modest role in pure mathematics, but is an enormously powerful device in economics. Definition 58 a set in rn is called compact if it is closed and bounded. Math 431 real analysis i solutions to homework due october 1. In metric spaces, a compact space glorified set math. A set is closed if it contains the limit of any convergent sequence within it. A of open sets is called an open cover of x if every x.740 780 157 483 1205 985 1598 442 678 981 115 1562 671 676 1049 740 430 977 672 727 1529 87 581 1166 661 674 1102 892 563 1135