theorem Th55: :: TSEP_1:55

for X being non empty TopSpace
for A1, A2 being Subset of X st A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )