1. Suppose for contradiction that A is not connected. Then you can write it as the union of 2 non-empty separated sets.
2. So A^C is not connected. Now suppose is not connected. Then you can write it as the union of 2 non empty separable sets.
1) Let (X, d) be a metric space and let A, B be two closed subsets of X such that the union and intersection of A and B are connected. Prove that A is connected.
2) Let (X, d) be a connected metric space and let A be a connected subset of X. Assume that the complement of A is the union of two separated sets B and C. Prove that the union of A and B are connected.
Thanks in advance!
Suppose is not connected. We know that is a separation of . Let be a separation of . Then , , and . Now . Now is completely contained in either or . WLOG suppose . Then . Also . Then . Also . So and are closed. Hence which is a separation of . Contradiction.