
theorem
  for T be T_0 TopSpace st T is infinite for B be Basis of T holds B is
  infinite
proof
  let T be T_0 TopSpace;
  assume
A1: T is infinite;
  let B be Basis of T;
  assume B is finite;
  then reconsider B1 = B as finite Subset-Family of T;
  union Components(B1) = the carrier of T by Th15;
  then consider X be set such that
A2: X in Components(B1) and
A3: X is infinite by A1,FINSET_1:7;
  reconsider X as infinite set by A3;
  consider x be object such that
A4: x in X by XBOOLE_0:def 1;
  consider y be object such that
A5: y in X and
A6: x <> y by A4,SUBSET_1:48;
  reconsider x1 = x, y1 = y as Element of T by A2,A4,A5;
  consider Y be Subset of T such that
A7: Y is open and
A8: x1 in Y & not y1 in Y or y1 in Y & not x1 in Y by A1,A6,T_0TOPSP:def 7;
  now
    per cases by A8;
    suppose
A9:   x in Y & not y in Y;
      then x in Y /\ X by A4,XBOOLE_0:def 4;
      then X c= Y by A2,A7,Th26,XBOOLE_0:4;
      hence contradiction by A5,A9;
    end;
    suppose
A10:  y in Y & not x in Y;
      then y in Y /\ X by A5,XBOOLE_0:def 4;
      then X c= Y by A2,A7,Th26,XBOOLE_0:4;
      hence contradiction by A4,A10;
    end;
  end;
  hence contradiction;
end;
