Appendix FPath ConnectednessIn order to avoid having to define a general topological space, we shall phrasethis appendix in terms of metric spaces. However, the reader should be awarethat this material is far more general than we are presenting it. We assume thatthe reader has studied Appendix D.In elementary analysis and geometry, one thinks of a curve as a collectionof points whose coordinates are continuous functions of a real variable l. Forexampie, 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 parametert. If we require that the curve join two points p and q, then the parameter canalwaysbe adjustedsothat l:0 atp and f:1 at g. Thusweseethat thecurveis described by a continuous rnapping from the unit interval I : [0,1] into theplane.Let X be a metric space, and let 1 - [0, 1] be a subspace of lR with theusual metric. We define a path in X, joining two points p and q of X, to beacontinuousmapping f : I - X such that /(0) :pand /(1) :q. Thispathwill be said to lie in a subset A c X if f (I) c A. It is iniportant to realize thatthe path i,s the mappi,ng /, and not the set of image points /(1). The space Xis said to be path connected if for every p,q € X there exists a path in Xjoining p and q. lf A c X , then .4 is path connected if every pair of points of Acan be joined by a path in ,4. (We should note that what we have called patliconnected is sometimes called arcwise connected.)Let us consider for a moment the space IR". If ri,z; € lR', then welet r;4-denote the closed line segment joining ri and r;. A subset ,4 C lR" is said tobe polygonally connected if given any two points p, q e ,4 there are pointsro: p, 11, fr2, . . ., tm: q in A such that Wrr.i-lri C A.r47r42APPENDIX F. PATH CONNECTEDNESS.4 c IR2Just because a subset of IR" is path connected does not mean that it ispolygonally connectecl. For example, the unit circle in lR2 is path connectedsince it is actually a path itself, but it is not polygonally connected.Example F.1. The space lR' is path connected. Indeed, if p € IR" has coordi-nates (o1, ...,fr") and g € R'has coordinates (ut,.'.,gn), then we define themapping f : I +' Wn br/(r) : (.fl(t),...,.f"(r)) where Jt(t) : (1 - t)il + 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 pathconnected,,The following is a simple consequence of Theorem D.5 that we shall need forour main result (i.e., Theorem F.2).Theorem F.L. Let ! t (X*d4) - (X2,d2) and,9: (Xz,d2)'-', (Xs,fu) both becontinuous funct'ions. Then go f : (X1,d'1) - (Xs,d,s) i's a continuous functi'on.Proof. If U c Xs is open, then the continuity of g shows that 9-1(U) C Xz isopen. Therefore (s o f)-r(rJ) : (/-t " s-L)(U) : f -r(g-r(U)) is open by thecontinuity of /.Theorem F,2. Let f be a continuous mapp'ing frorn a metric space X onto ametric spaceY. ThenY is path connected,i'f X i's.Proof . Let r' , y' be any two points of Y. Then (since / is surjective) there existn,g e X such that f ("): r/ and f (a): g'. Since X is path connected, thereexists apath gjoining r andg such that 9(0): r andg(1): g. But then /ogis a continuous function (Theorem F.1) from 1 into Y such that (/ o 9)(0) : r'and (/o9)(1): g/. Inotherwords, /og is apathjoining x'andg', andhenceIIY is path connected.143It is an obvious corollary of Tireorern F.2 that if / is a contirnrous nrappingfronr the pat,h connected space X 'into }/, then /(X) is path connected in Ysince / maps X onto the subspace