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2. GEOMETRIC PROPERTIES

FIGURE 2.1

Let "Y be the open segment (z 1 ,z 2 ) and suppose that "Yn8D = 0. Then

"YU f("Y) U {z1, z2}

is a Jordan curve which bounds a domain G C B(z 0 , c:). By our assumptions about

f, f(G) = G.

If w 1 , w 2 are points in G n D, then w 1 and w2 can be joined by an arc a in G

and hence by the continuum

(3 =(an D) u f(a n D*) c G n D.

Thus G n D is connected. The same is true of G n D• and we conclude that

F1 = GnD, F2 = GnD·

are closed, connected sets with

Since R

2

\ (F 1 U F 2 ) is a Jordan domain and hence connected, E = F 1 n F 2 is a

connected set which joins z 1 and z 2 in f)D n B lzo, c:). See, for example, Theorem

V .11.5 in Newman [140].

Next if "Y n 8D =f. 0, then

"Y = (Uj"'fj) U ("Y n 8D)

where "Yj is an open segment (z 1 j, z 2 j ) with z 1 j, Z2jo E 8D and "Yj n 8D = 0. Then

as above there exists a connected set Ej which joins z 1 j and Z2j in f)D n B(zo, c:)

and

E = (UjEj) U ("Y n 8D)

is a connected set which joins z 1 and z 2 in f)D n B (z 0 , c:). Thus f)D is locally

connected at z 0. Since z 0 was chosen arbitrarily, f)D itself is locally connected. 0