Math 446646 Important facts about Topological Spaces, Part II Summary: Math 446­646 Important facts about Topological Spaces, Part II Connected sets · Let (X, T) be a topological space. Assume that there are two non-empty open subsets A, B of X such that X = A B and A B = . Then {A, B} is a partition of X. · Notice that if {A, B} is a partition of X, the sets A and B are both open and closed. · A topological space (X, T) is connected if it does not admit any partition. · Some characterizations of connectedness: (a) (X, T) is connected if and only if the only subsets of X that are both open and closed are X and . (b) (X, T) is connected if and only if every continuous function f : X {0, 1} is constant. · Let (X, T) be a topological space, let {A}I be a collection of subsets of X such that each (A, TA ) is connected. Assume that IA is non-empty. Then IA is connected. · Let (X, T) be a topological space, let A be a subset of X such that (A, TA) is connected. Then any set B such that A B A is connected. · Let (X, T) be connected and f : (X, T) (Y, T ) be continuous and onto. Collections: Mathematics