Topology Connectedness Question
Is it true, given A,B as two connected subsets of a topological space X s.t.
A intersection closure(B) is non-empty, that:
If C is an open & closed subset of AUB, then CU((AUB)\C)=AUB
I am not sure as both C & ((AUB)\C) are open, so is boundary(C) in either of these open sets?
If not, can you please suggest any alternative routes to tackle this problem:
Given A,B as two connected subsets of a topological space X s.t.
A intersection closure(B) is non-empty, prove that AUB is connected.
(I have tried to show that the empty set and AUB are the only two open & closed subsets of AUB by contradiction (supposing there exists C with this property))
Any help would be much appreciated,