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 G being Subset of X st G <> {} & G is open holds
  ( A`) meets G
proof
  thus A is boundary implies for G being Subset of X st G <> {} & G is open
  holds ( A`) meets G
  by SUBSET_1:24,TOPS_1:50;
  assume
A1: for G being Subset of X st G <> {} & G is open holds ( A`) meets G;
  assume Int A <> {};
  then Int A c= A & ( A`) meets Int A by A1,TOPS_1:16;
  hence contradiction by SUBSET_1:24;
end;
