1 *10 I. This exercise is used to prove *Corollary 11.29.

(a) A set E ~ Rn is said to be polygonally connected if and only if any two points a,b E E can be connected by a polygonal path in E; i.e., there exist points Xk E E, k = 1, ... , N, such that Xo =

a, XN = band L(Xk-l;Xk) ~ E for k = 1, ... , N. Prove that every polygonally connected set in Rn is connected.

(b) Let E ~ R n be open and Xo E E. Let U be the set of points x E E that can be polygonally connected in E to Xo. Prove that U is open.

(c) Prove that every open connected set in Rn is polygonally connected.