let T1, T2 be TopSpace; :: thesis: ( T1 is second-countable & T1,T2 are_homeomorphic implies T2 is second-countable )
assume T1 is second-countable ; :: thesis: ( not T1,T2 are_homeomorphic or T2 is second-countable )
then A1: weight T1 c= omega by WAYBEL23:def 6;
assume T1,T2 are_homeomorphic ; :: thesis: T2 is second-countable
then weight T1 = weight T2 by METRIZTS:4;
hence T2 is second-countable by A1, WAYBEL23:def 6; :: thesis: verum