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## ANALYSIS I

by: Cassidy Grimes

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2

# ANALYSIS I MATH 554

Cassidy Grimes

GPA 3.51

R. Sharpley

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COURSE
PROF.
R. Sharpley
TYPE
Class Notes
PAGES
2
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 554 at University of South Carolina - Columbia taught by R. Sharpley in Fall. Since its upload, it has received 10 views. For similar materials see /class/229529/math-554-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
MATH 554703 1 FALL 08 Lecture Note Set 3 Introduction to Metric Space Topology Please read Chapter 2 of Rudin which deals with the concepts of metric space topology pages 30 36 Some portions are outlined again here in the order of lecture presentation for your convenience An idea of the proof is given in some cases but the full proofs are given during in class Defn A set X equipped with a non negative function d X x X gt 0 00 is called a metric space if d satis es 1 dpq Z 0 for allpq E X and dpq 0 if and only ifp q 2 619961 dam 3 109603 611 r drq for any r E X Defn A neighborhood of radius r of a point p is de ned by NAP I q E XI 10961 lt 7 On the real line R the neighborhoods of a point a take the form of intervals a r a r since the inequality a x lt r is equivalent to r lt x a lt r which in turn is equivalent to a r lt x lt a r Defn Suppose X7d is a metric space 1 A limit point p of a set E Q X is a point for which each of its neighborhoods contains a point from E distinct from p 2 E denotes the set of all limit points of a set E and is called its derived set 3 A set E is called closed if it contains all its limit points ie E Q 4 E z E U E denotes the closure of a set E 5 A point p of E is called an isolated point if it is not a limit point of E Theorem If p is a limit point of E then each neighborhood of p contains an in nite number of members of E think of the proof that every interval contains an in nite number of rational numbers7 from the result that each interval had at least one Corollary A nite set has no limit points ie 7 E nite implies E Defn Suppose X7d is a metric space 1 A point p is called an interior point of E if there exists an r gt 0 so that NAP Q E 2 A set E is called open if each point of E is an interior point of E 3 The set of all interior points is called the interior of E and is denoted by E0 In this case a set E is open is equivalent to E Q E0 Theorem In a metric space each neighborhood is an open set Proof For q E BT1007 let 6 r 7 doo7q7 then B5q C BTp0 Corollary For any set E 7 the interior of E is an open set Proof lfp 6 E07 then there is a nhbd NTp Q E From the previous theorem NTp is open and so contains a corresponding nhbd for each of its points This shows that each point of NTp is an interior point of Theorem A set E is open if and only if its complement EC is closed Essentially a tautology Using the de nitions each point of E is an interior point ltgt each point of E has a nhbd with no elements of E0 ltgt no element of E can be a limit point of E0 ltgt E0 contains all its limit points Corollary Let X d be a metric space then 1 Arbitrary unions of open sets are open Use an r from any of the open sets of the collection to which the point belongs 2 Finite intersections of open sets are open Use the smallest r for the open sets of the collection to which the point belongs 3 Arbitrary intersections of closed sets are closed Use De Morgan7s law and apply the previous Theorem 4 Finite unions of closed sets are closed Use De Morgan7s law and apply the previous Theorem Note Arbitrary intersectons of open sets need not be open a lfOn ln71n then a 9n nl b If on ln 1 171 then 9 01 nl c If on ln 1 then 9 01 nl Examples examples examples For example7 R7 7 RT discrete metric7 Ca7 b the space of continuous functions

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