
theorem Th25:
  for X being non empty set, P being Subset of CofinTop X holds P
  is closed iff P = X or P is finite
proof
  let X be non empty set, P be Subset of CofinTop X;
  set T = CofinTop X;
  hereby
    assume that
A1: P is closed and
A2: P <> X;
    P` in the topology of T by A1,PRE_TOPC:def 2;
    then P in COMPLEMENT the topology of T by SETFAM_1:def 7;
    then
A3: P in {X} \/ Fin X by Def6;
    not P in {X} by A2,TARSKI:def 1;
    then P in Fin X by A3,XBOOLE_0:def 3;
    hence P is finite;
  end;
  assume
A4: P = X or P is finite;
  the carrier of T = X by Def6;
  then P in {X} or P in Fin X by A4,FINSUB_1:def 5,TARSKI:def 1;
  then P in {X} \/ Fin X by XBOOLE_0:def 3;
  then P in COMPLEMENT the topology of T by Def6;
  then P` in the topology of T by SETFAM_1:def 7;
  then P` is open;
  hence thesis;
end;
