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

theorem
  T is second-countable implies ex B being Basis of T st B is countable
proof
  set B = the Basis of T;
  consider B1 being Basis of T such that
  B1 c= B and
A1: card B1 = weight T by TOPGEN_2:18;
  assume T is second-countable;
  then card B1 c= omega by A1,WAYBEL23:def 6;
  then card card B1 c= card omega;
  then B1 is countable;
  hence thesis;
end;
