let T1, T2 be TopSpace; :: thesis: ( the carrier of T1 = the carrier of T2 & ( for A1 being Subset of T1
for A2 being Subset of T2 st A1 = A2 holds
Cl A1 = Cl A2 ) implies the topology of T1 = the topology of T2 )
assume that
A1: the carrier of T1 = the carrier of T2 and
A2: for A1 being Subset of T1
for A2 being Subset of T2 st A1 = A2 holds
Cl A1 = Cl A2 ; :: thesis: the topology of T1 = the topology of T2
now :: thesis: for A being set holds
( ( A is closed Subset of T1 implies A is closed Subset of T2 ) & ( A is closed Subset of T2 implies A is closed Subset of T1 ) )
let A be set ; :: thesis: ( ( A is closed Subset of T1 implies A is closed Subset of T2 ) & ( A is closed Subset of T2 implies A is closed Subset of T1 ) )
thus ( A is closed Subset of T1 implies A is closed Subset of T2 ) :: thesis: ( A is closed Subset of T2 implies A is closed Subset of T1 )
proof
assume A is closed Subset of T1 ; :: thesis: A is closed Subset of T2
then reconsider A1 = A as closed Subset of T1 ;
reconsider A2 = A1 as Subset of T2 by A1;
Cl A1 = A1 by PRE_TOPC:22;
then Cl A2 = A2 by A2;
hence A is closed Subset of T2 ; :: thesis: verum
end;
assume A is closed Subset of T2 ; :: thesis: A is closed Subset of T1
then reconsider A2 = A as closed Subset of T2 ;
reconsider A1 = A2 as Subset of T1 by A1;
Cl A2 = A2 by PRE_TOPC:22;
then Cl A1 = A1 by A2;
hence A is closed Subset of T1 ; :: thesis: verum
end;
then TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #) by Th6;
hence the topology of T1 = the topology of T2 ; :: thesis: verum