reserve a,b,c for set;

theorem
  for X,x0 being set st X is infinite for U being Subset of
  DiscrWithInfin(X,x0) st U = {x0} holds U is not open
proof
  let X,x0 be set;
  set T = DiscrWithInfin(X,x0);
  assume
A1: X is infinite;
  let U be Subset of DiscrWithInfin(X,x0);
  assume
A2: U = {x0};
  the carrier of T = X by Def5;
  then X = U` \/ {x0} by A2,XBOOLE_1:45;
  then
A3: U` is infinite by A1;
  x0 in U by A2,TARSKI:def 1;
  hence thesis by A3,Th19;
end;
