let T be TopSpace; :: thesis: for A being Subset of T holds
( A is 1st_class iff Border A is empty )
let A be Subset of T; :: thesis: ( A is 1st_class iff Border A is empty )
A1: ( Border A is empty implies A is 1st_class )
proof
assume Border A is empty ; :: thesis: A is 1st_class
then (Int (Cl A)) \ (Cl (Int A)) = {} by Th21;
then Int (Cl A) c= Cl (Int A) by XBOOLE_1:37;
hence A is 1st_class ; :: thesis: verum
end;
( A is 1st_class implies Border A is empty )
proof
assume A is 1st_class ; :: thesis: Border A is empty
then Int (Cl A) c= Cl (Int A) ;
then (Int (Cl A)) \ (Cl (Int A)) = {} by XBOOLE_1:37;
hence Border A is empty by Th21; :: thesis: verum
end;
hence ( A is 1st_class iff Border A is empty ) by A1; :: thesis: verum