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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.
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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).