In: Advanced Math
(Connected Spaces)
(a) Let <X, d> be a metric space and E ⊆ X. Show that E is
connected iff for all p, q ∈ E, there is a connected A ⊆ E with p,
q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k is connected.