theorem Th39: :: WAYBEL14:39

for L being complete Scott TopLattice holds
( ( for x being Element of L ex B being Basis of x st
for Y being Subset of L st Y in B holds
( Y is open & Y is filtered ) ) iff for V being Element of (InclPoset (sigma L)) ex VV being Subset of (InclPoset (sigma L)) st
( V = sup VV & ( for W being Element of (InclPoset (sigma L)) st W in VV holds
W is co-prime ) ) )