theorem Th9: :: TOPGEN_3:9

for X being set
for i being Function of (bool X),(bool X) st i . X = X & ( for A being Subset of X holds i . A c= A ) & ( for A, B being Subset of X holds i . (A /\ B) = (i . A) /\ (i . B) ) & ( for A being Subset of X holds i . (i . A) = i . A ) holds
for T being TopStruct st the carrier of T = X & the topology of T = rng i holds
( T is TopSpace & ( for A being Subset of T holds Int A = i . A ) )