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