1 *10 I. This exercise is used to prove *Corollary 11.35. (a) A set E ...
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1 *10 I. This exercise is used to prove *Corollary 11.35.
(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 = b and 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.
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