reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;

theorem Th9:
  T1 is second-countable & T1,T2 are_homeomorphic
  implies T2 is second-countable
proof
  assume T1 is second-countable;
  then
A1: weight T1 c= omega by WAYBEL23:def 6;
  assume T1,T2 are_homeomorphic;
  then weight T1 = weight T2 by METRIZTS:4;
  hence T2 is second-countable by A1,WAYBEL23:def 6;
end;
