reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  x for set;

theorem Th54:
  IRRAT is boundary Subset of R^1
proof
  A` is dense by Lm2,Th51,TOPMETR:17;
  hence thesis by TOPS_1:def 4;
end;
