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,

TMH