theorem
  P is boundary iff for Q st Q c= P & Q is open holds Q = {}
proof
  hereby
    assume P is boundary;
    then
A1: P` is dense;
    let Q;
    assume that
A2: Q c= P and
A3: Q is open and
A4: Q <> {};
    Q meets P` by A1,A3,A4,Th45;
    hence contradiction by A2,SUBSET_1:24;
  end;
  assume
A5: for Q st Q c= P & Q is open holds Q={};
  assume not P is boundary;
  then not P` is dense;
  then consider C being Subset of TS such that
A6: C <> {} and
A7: C is open and
A8: (P`) misses C by Th45;
  C c= P by A8,SUBSET_1:24;
  hence contradiction by A5,A6,A7;
end;
