Connectedness When to Use Use this skill when working on connectedness problems in topology. Decision Tree 1. Is X connected? - Strategy 1 - Contradiction: Assume X = U union V where U, V are disjoint, non-empty, and open Derive a contradiction - Strategy 2 - Path connectedness: Show for all x,y in X, exists continuous path f: [0,1] - X with f(0)=x, f(1)=y - Strategy 3 - Fan lemma: If {A i} are connected sharing a common point, then union A i is connected 2. Connectedness Proofs - Show no separation exists - - Use intermediate value theorem for R subsets 3. Path Connectedness - Construct expl…