By Graeme L. Cohen

Designed for one-semester classes for senior undergraduates, this booklet methods subject matters firstly via convergence of sequences in metric house. notwithstanding, the choice topological process can be defined. purposes are integrated from differential and necessary equations, platforms of linear algebraic equations, approximation thought, numerical research and quantum mechanics.

Cover; Half-title; Series-title; name; Copyright; Contents; Preface; 1 Prelude to trendy research; 2 Metric areas; three The fastened element Theorem and its functions; four Compactness; five Topological areas; 6 Normed Vector areas; 7 Mappings on Normed areas; eight internal Product areas; nine Hilbert area; Bibliography; chosen recommendations; Index.

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Sample text

And it is clear with this arrangement how the integers may be counted. It now follows that any other set is countable if it can be shown to be equivalent to Z. In fact, any countable set may be used in this way to prove that other sets are countable. The next theorem gives two important results which will cover most of our applications. The second uses a further extension of the notion of a union of sets, this time to a countable number of sets: if X±, X 2, . , are sets, then OO |^J Xk = {x : x G Xk for at least one k = 1, 2, 3, .

In particular, of course, all closed intervals are compact subsets of R . 6 (1) Show that open intervals are open sets and closed intervals are closed sets. (2) Let S be a point set. (a) A point x E R is an interior point of S if, for some <5 > 0, (x — 8, x + 8) C S. Show that all points of R are interior points, (b) A point x E R is an isolated point of S if, for some 8 > 0, (x — 8, x + 8) H S = {x } . Show that all points of Z are isolated points. (3) Let S be a point set. A point x E R is a boundary point of S if, for every 8 > 0, (x —8, x + 8) (l S ^ 0 and (x —8, x + (5) H ~ S ^ 0 .

Show that A U B is bounded above, and sup A U B = max {sup A, sup B }. (4) (a) Show directly that x ^ P ln L ilA V 72]? f°r any positive real number x. (b) Show that Pl^L-^O, 1/n) = 0 . 6 O p e n and clo se d sets Topology is a branch of mathematics dealing with entities called open sets. Their properties are modelled on those defined below for real numbers. These help us with a further investigation of real numbers, including the notion o f compactness of subsets of R . The work in this section is sometimes called topology of the real line.

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