Please see the attached file for the fully formatted problems.
Let A and B be separated subsets of some Rk, suppose and , and define
for . Put , . [Thus if and only if .]
(a) Prove that Ao and Bo are separated subsets of R1 .
(b) Prove that there exists such that .
(c) Prove that every convex subset of Rk is connected
Connected convex subsets are investigated.