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Transcript
Appendix F Path Connectedness In order to avoid having to define a general topological space, we shall phrase this appendix in terms of metric spaces. However, the reader should be aware that this material is far more general than we are presenting it. We assume that the reader has studied Appendix D. In elementary analysis and geometry, one thinks of a curve as a collection of points whose coordinates are continuous functions of a real variable l. For exampie, a curve in the plane IR2 may be specified by giving its coordinates @ : f(t),A : gft)) where / and g are continuous functions of the parameter t. If we require that the curve join two points p and q, then the parameter can at g. Thusweseethat thecurve alway s be adjus t eds ot hat l: 0 atp and f:1 is described by a continuous rnapping from the unit interval I : [0,1] into the plane. Let X be a metric space, and let 1 - [0, 1] be a subspace of lR with the usual metric. We define a path in X, joining two points p and q of X, to be ac ont inuous m apping f : I X such that /(0) :pand /(1) :q. Thispath will be said to lie in a subset A c X if f (I) c A. It is iniportant to realize that the path i,s the mappi,ng /, and not the set of image points /(1). The space X is said to be path connected if for every p,q € X there exists a path in X joining p and q. lf A c X , then .4 is path connected if every pair of points of A can be joined by a path in ,4. (We should note that what we have called patli connected is sometimes called arcwise connected.) Let us consider for a moment the space IR". If ri,z; € lR', then welet r;4denote the closed line segment joining ri and r;. A subset ,4 C lR" is said to be polygonally connected if given any two points p, q e ,4 there are points r o: p, 11, f r 2, . . . , t m : q in A s u c h t h a t Wr r . i - l r i C A. r47 r42 APPENDIX F. PATH CONNECTEDNESS .4 c IR2 Just becausea subsetof IR" is path connecteddoes not mean that it is polygonally connectecl.For example,the unit circle in lR2 is path connected sinceit is actually a path itself, but it is not polygonallyconnected. Example F.1. The spacelR' is path connected.Indeed, if p € IR" has coordinates(o1,...,fr") and g € R'has coordinates(ut,.'.,gn), then we definethe m a p p i n gf : I + ' W nbr/(r) : (.fl (t),...,.f" (r)) w hereJt(t) : (1 - t)i l + tai . This mapping is clearly continuous and eatisfies/(0) : p and /(1) _: q. Thus / is a path joiniug the arbitrary points p and g of R', and hence IRt is path connected,, of TheoremD.5 that we shall needfor The followingis a simpleconsequence our main result (i.e., TheoremF.2). (X2,d2) and,9: (Xz,d2)'-', (Xs,fu) bothbe Theorem F.L. Let ! t (X*d4) continuousfunct'ions. Thengo f : (X1,d'1) (Xs,d,s)i'sa continuousfuncti'on. Proof. If U c Xs is open, then the continuity of g showsthat 9-1(U) C Xz is -r(g-r(U)) is openby the : open. Therefore(s o f)-r(rJ) : (/-t " s-L)(U) f I continuity of /. Theorem F,2. Let f be a continuous mapp'ing frorn a metric space X onto a metric spaceY. ThenY is path connected,i'fX i's. Proof . Let r' , y' be any two points of Y. Then (since / is surjective) there exist n,g e X such that f ("): r/ and f (a): g'. Since X is path connected,there ex is t s apat h gjoinin g r a n d g s u c h t h a t 9 ( 0 ) : r a n d g ( 1 ) : g . B u t t h e n / o g is a continuous function (Theorem F.1) from 1 into Y such that (/ o 9)(0) : r' g/ . I n o t h e r w o r d s , / o g i s a p a t h j o i n i n g x 'a n d g ', a n d h e n c e and ( / o9) ( 1) : I Y is path connected. 143 It is an obvious corollary of Tireorern F.2 that if / is a contirnrous nrapping fronr the pat,h connected space X 'into }/, then /(X) is path connected in Y since / maps X onto the subspace/(X).