reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;

theorem
  A is boundary iff for C being Subset of X st A` c= C & C is closed
  holds C = the carrier of X
proof
  thus A is boundary implies for C being Subset of X st A` c= C & C is closed
  holds C = the carrier of X
  proof
    assume
A1: A is boundary;
    let C be Subset of X;
    assume A` c= C;
    then
A2: C` c= A`` by SUBSET_1:12;
    assume C is closed;
    then C` = {}X by A1,A2,TOPS_1:50;
    hence thesis by Th2;
  end;
  assume
A3: for C being Subset of X st A` c= C & C is closed holds C = the
  carrier of X;
  now
    let G be Subset of X;
    assume that
A4: G c= A and
A5: G is open;
    A` c= G` by A4,SUBSET_1:12;
    then G` = the carrier of X by A3,A5;
    hence G = {} by Th1;
  end;
  hence thesis by TOPS_1:50;
end;
