Appendix DMetric SpacesFor those readers not already familiar with the elementary properties of met-ric spaces and the notion of compactness, this appendix presents a sufficientlydetailed treatment for a reasonable understanding of this subject matter. How-ever, for those who have already had some exposure to elementary point settopology (or even a solid introduction to real analysis), then the material inthis appendix should serve as a useful review of some basic concepts. Besides,any mathematics or physics student should become thoroughly familiar with allof this material.D.l Definitions and Open SetsLet S be any set. Then a function d: S x S - IR is said to be a metric on Sif it has the following properties for all r, A,z € S:(M1) d(r,s) > 0;(M2) d(r,s): 0 if and only 1f r : s;(M3) d(r, a) : d(a,r);(Ma) d@,a) + d(a, z) > d(r, z).The real number d(r,g) is called the distance between x and y, and the set Stogether with a metric d is called a metric space (S,d).As a simple example, let S : lR and let d(r,g) : lt - gl for all r,g € IR.From the properties of the absolute value, conditions (M1)-(M3) should beobvious, and (M4) follows by simply noting thatlr - z1 : lr - a * a - zl < l" - ul + la - tl.For our purposes, we point out that given any normed vector space (% ll ' ll)we may treat V as a metric space by definingd(n,s):ll*-sll101t02APPENDIX D, METHIC SPACESfor every r,g € V. Using Theorem ??, the reader should have no troubleshowing that this does indeed define a metric space (y'd). In fact, it is easy tosee that lR.' forms a metric space relative to the standard inner product and itsassociated norm.Given a metric space (X,d) and any real number r ) 0, the open ball ofradius r and center re is the set Ba(zo, r) c X defined byBa@s,r) : {r € X : d(r,ro) < r}.Since the metric d is usually understood, we will generally leave off the subscriptd and simply write B(rs,r). Such a set is frequently referred to as an r-ball.We say that a subset (l of X is open if, given any point r € U, there existsr > 0 and an open ball B(r,r) such that B(r,r) c U.Probably the most common example of an open set is the open unit disk D1' lht r - r 1tn K- oeflneo DvDt -- {(r,g) e R2 , *" + 92 < lY.We see that given any point ro € D1, we can find an open bail B(rg,r) C D1 bychoosing r : 1, -d(re, 0) (see the picture in the proof of the following theorem).The setD2 : {(r,y) e R' 12 + y' < L}is not open because there is no open ball centered on any of the boundary points12 + A2 : 1 that is contained entirely within Dz.The fundamental characterizations of open sets are contained in the followingthree theorems.Theorem D.L. Let (X,d) be a metric space. Then anu open ball is an openset.Proof. Let B(rs,r) be an open bail in X and let u be any point in B(rs,r).We musL find a B(r, r') contained in B(rs. r).B(16,r)D.1. DEFINITIO-IVS AAID OPEAI SETSSince d(r, rs) 1r, we deflne rt : r - d(r,ts)' Then for any A e B(t,r') wehave d(y,r) < r' and henced(a,ril < d(a,x) * d(r,rg) 1r' I d(r,16) : rwhich shows that 9 € B(rg,r). Therefore B(t,r') C B(ts,r). !Theorem D.2. Let (X,d,) be a metric space. Then(i) Both X and, o are open sets.(ii) The intersection of a finite number of open sets is open.(i,ii) The uni,on of an arbitrarg number of open sets is open.Proof. (i) X is clearlyopensince for anyr € X andr ) 0 wehave B(t,r) c X.The statement tha| a is open is also automatically satisfied since for any r € a(there are none) and r > 0, we again have B(r,r) c a.(ii) Let {(I},i e 1be a finite collection of open sets in X. Suppose {4}is empty. Then nt/i : X because a point is in the intersection of a collectionof sets if it belongs to each set in the collection, so if there are no sets in thecollection, then every point of X satisfies this requirement. Hence nUi : y i"open by (i).Now assume {U.} It not empty and let U : )U;.. lf U : O then it is openby (i), so assume U + a. Suppose r eU sothat r € U; for every i e 1. Thenthere exists B(r,r1) C [J.; for each i, and since there are only a finite number ofthe ri we may let r: min{rl}. It follows thatB(r,r) C B(r,r;) cUifor every i, and hence B(r,r) c aui : U. In other words, we have found anopen ball centered at each point of U and contained in U, thus proving that Uis open.(iii) Let {U,} b" an arbitrary collection of open sets. If {U,} it empty, thenU : uUr: a is open by (i). Now suppose {t/,} is not empty and r € t-)Ui.Then z € Ut for some i, and hence there exists B(r,r1) C Ut C UU' so that uU,is open. !Notice that part (ii) ofthis theorem requires that the collection be finite. Tosee the necessity of this, consider the infinite collection of intervals in JR givenby (-lln,Iln) for 1/-n <. oo. The intersection of these sets is the point {0}which is not open in IR.In an arbitrary metric space the structure of the open sets can be verycomplicated. However, the most general description of an open set is containedin the following.Theorem D.3.,4 subset U of a rnetric space (X,d') is open if and only if it isthe un'ion of open balls.103104APPEI,{DIX D. METRIC SPACESProol. Assume U is the union of open bails. By Theorem D.l each open ballis an open set, and hence U is open by Theorem D.2(iii). Conversely, let U bean open subset of X. For each r € [/ there exists at least one B(t,r) C U sothat U,EyB(r,r) C U. On the other hand each r € [/ is contained in at leastB(r,r) so that (l cu,€uB(r,r). ThereforeU -l,EuB(r,r). tlAs a passing remark, note that a set is never open in and of itself. Rather,a set is open only with respect to a specific metric space containing it. Forexample, the set of nurnbers [0, 1) is not open when considered as a subset ofthe real line because any open interval about the point 0 contains points not in[0, 1). However, if [0, 1) is considered to be the entire space X, then it is openby Theorem D.2(i).If t/ is an open subset of a rnetric space (X,d), then its complemett U' :X - U is said to be closed. In other words, a set is closed if and only if itsconiplement is open. For exarnple, a mornents thought should convince you thatthe srrbset of lR.z defined by {(r,y) e R' , 12 + a2 < 1} is a closed set. Theclosed ball of radius r centered at 16 is the set B[rs,r) defined in the obviousway byBfro,rl: {z e X : d(rs,") < r}.We leave it to the reader …