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

theorem Th10: :: also for empty TopSpaces
  for T being discrete non empty TopSpace holds T is separable iff
  card [#]T c= omega
proof
  let T be discrete non empty TopSpace;
  hereby
    assume T is separable;
    then
A1: density T c= omega by TOPGEN_1:def 13;
    ex A being Subset of T st A is dense & density T = card A by
TOPGEN_1:def 12;
    hence card [#]T c= omega by A1,TOPS_3:19;
  end;
  assume card [#]T c= omega;
  then T is countable by TOPGEN_1:2;
  hence thesis;
end;
