Question

If a topological space is a union if disjoint it is saud to be

Answer 1

Consider the topological space Z defined as the disjoint union Z=X∪Y. Please tell me if these statements are true:

1) If both X and Y are open in Z or both are closed in Z then Z must be disconnected

2) If one of them is open and the other is closed then Z may or may not be connected. For example {0} is closed in [0,1] and (0,1] is open in [0,1] but {0}∪(0,1]=[0,1] is connected. The second example, take Z=(0,1)∪[2,3] then (0,1) is open in Z and [2,3] is closed in Z but Z is not connected. Thanks alot.