theorem :: TOPGEN_3:3

for X being set
for B being non-empty ManySortedSet of X st rng B c= bool (bool X) & ( for x, U being set st x in X & U in B . x holds
x in U ) & ( for x, y, U being set st x in U & U in B . y & y in X holds
ex V being set st
( V in B . x & V c= U ) ) & ( for x, U1, U2 being set st x in X & U1 in B . x & U2 in B . x holds
ex U being set st
( U in B . x & U c= U1 /\ U2 ) ) holds
ex P being Subset-Family of X st
( P = Union B & ( for T being TopStruct st the carrier of T = X & the topology of T = UniCl P holds
( T is TopSpace & ( for T9 being non empty TopSpace st T9 = T holds
B is Neighborhood_System of T9 ) ) ) )