let S be non empty SubSpace of M; :: thesis: ( S is second-countable & S is Hausdorff )
[#] S c= [#] M by PRE_TOPC:def 4;
then reconsider CS = [#] S as Subset of M ;
consider B being Basis of (M | CS) such that
A1: card B = weight (M | CS) by WAYBEL23:74;
A2: weight (M | CS) c= omega by WAYBEL23:def 6;
A3: TopStruct(# the carrier of S, the topology of S #) = S | ([#] S) by TSEP_1:93;
A4: S | ([#] S) is SubSpace of M by TSEP_1:7;
[#] (S | ([#] S)) = [#] S by PRE_TOPC:def 5;
then TopStruct(# the carrier of S, the topology of S #) = M | CS by A3, A4, PRE_TOPC:def 5;
then B is Basis of S by YELLOW12:32;
then weight S c= card B by WAYBEL23:73;
then weight S c= omega by A1, A2;
hence ( S is second-countable & S is Hausdorff ) ; :: thesis: verum