reserve a,b,c for set;

theorem Th19:
  for X,x0 being set, A being Subset of DiscrWithInfin(X,x0) holds
  A is open iff not x0 in A or A` is finite
proof
  let X,x0 be set;
  set T = DiscrWithInfin(X,x0);
  set O1 = {U where U is Subset of X: not x0 in U};
  set O2 = {F` where F is Subset of X: F is finite};
  let A be Subset of T;
A1: the topology of T = O1 \/ O2 by Def5;
A2: the carrier of T = X by Def5;
  thus A is open implies not x0 in A or A` is finite
  proof
    assume A in the topology of T;
    then A in O1 or A in O2 by A1,XBOOLE_0:def 3;
    then (ex U being Subset of X st A = U & not x0 in U) or ex F being Subset
    of X st A = F` & F is finite;
    hence thesis by A2;
  end;
  assume not x0 in A or A` is finite;
  then A in O1 or A`` in O2 by A2;
  hence A in the topology of T by A1,XBOOLE_0:def 3;
end;
