Oct 15, 2012 ... If a < c < b then (c, b] ⊂ B0 and c has a nbdh (d, e) ⊂ B0 so d is an upper bound of A0 contradicting c = sup A0. Nathan Broaddus. General Topology and Knot Theory. Lecture 22 - 10/15/2012. Connectedness Components. Connectedness. Proof of Prop. 208 (continued). ▻ CASE II: Suppose c ∈ A0.