Connectedness

In Section 5.3 of Volume I we introduced the notion of connectedness of subsets of the real line, and showed that a non-empty subset of R is connected if and only if it is an interval. The notion extends easily to topological spaces. A topological space splits if X = F1 ⋃ F2, where F1 and F2 are disjoint non-empty closed subsets of (X, τ). The decomposition X = F1 ⋃ F2 is a splitting of X. If X does not split, it is connected. A subset A of (X, τ) is connected if it is connected as a topological subspace of (X, τ). If X = F1 ⋃ F2 is a splitting, then F1 = C(F2) and F2 = C(F1) are open sets, and so X is the disjoint union of two non-empty sets which are both open and closed; conversely if U is a non-empty proper open and closed subset of X, X = U ⋃ (X \ U) is a splitting of (X, τ). Thus (X, τ) is connected if and only if X and ø are the only subsets of X which are both open and closed.

Proposition 16.1.1Suppose that A is a connected subset of a topological space (X, τ) and that X = F1 ⋃ F2 is a splitting of X. Then either A ⊆ F1or A ⊆ F2.