Prove that E X is not connected if and only if there exist o

Prove that E X is not connected if and only if there exist open sets A, B X such that E A Union B, A Intersection B = theta), and E Intersection A and E Intersection B are both nonempty.

Solution

Let E be connected. Also, let there be disjoint/open sets A, B X , such that (EA)(EB)=E. This implies that E is a union of its two open subsets. This is a contradiction as E is connected. Hence E is not connected.

Conversely, if E is not equal to (EA)EB) for any two disjoint/open sets A and B which are subsets of X, then it implies that we also cannot express E as a union of any two of its open subsets , as, by the definition of a topological subspace, (EA) and(EB) together comprise the entire topological subspace E.