
theorem Th22:
  for T being TopSpace, K being Subset-Family of T
  holds the topology of T = UniCl K iff K is Basis of T
proof
  let T be TopSpace, K be Subset-Family of T;
  thus the topology of T = UniCl K implies K is Basis of T
  by CANTOR_1:1,CANTOR_1:def 2,TOPS_2:64;
  assume
A1: K is Basis of T;
  then K c= the topology of T by TOPS_2:64;
  then
A2: UniCl K c= UniCl the topology of T by CANTOR_1:9;
  the topology of T c= UniCl K by A1,CANTOR_1:def 2;
  hence the topology of T c= UniCl K & UniCl K c= the topology of T
  by A2,CANTOR_1:6;
end;
