a) A set E ⊂ Rn is said to be polygonally connected if and only if any two points a, b ∈ E can be connected by a polygonal path in E; that is, there exist points xk ∈ E, k = 1,. . . . . N, such that x0 = a, xN = b and L(xk-1; xk) ⊂ E for k = 1,..., N. Prove that every polygonally connected set in R" is connected.
b) Let E ⊂ Rn be open and x0 ∈ E. Let U be the set of points x ∈ E which can be polygonally connected in E to x0. Prove that U is open.
c) Prove that every open connected set in Rn is polygonally connected.