
theorem
  for T being TopSpace, P being Basis of T holds P is basis of T
proof
  let T be TopSpace, P be Basis of T;
A1: P c= the topology of T by TOPS_2:64;
  let p be Point of T, A be Subset of T such that
A2: p in Int A;
  the topology of T c= UniCl P & Int A in the topology of T by CANTOR_1:def 2
,PRE_TOPC:def 2;
  then consider Y being Subset-Family of T such that
A3: Y c= P and
A4: Int A = union Y by CANTOR_1:def 1;
  reconsider Y as Subset-Family of T;
  consider K being set such that
A5: p in K and
A6: K in Y by A2,A4,TARSKI:def 4;
  reconsider K as Subset of T by A6;
  take K;
  thus K in P by A3,A6;
  then K is open by A1;
  hence p in Int K by A5,TOPS_1:23;
A7: Int A c= A by TOPS_1:16;
  K c= union Y by A6,ZFMISC_1:74;
  hence thesis by A4,A7;
end;
