reserve a,b,c for set;
reserve r for Real,
  X for set,
  n for Element of NAT;

theorem Th38:
  for X being non empty set, x0 being Element of X for A being non
  empty Subset of x0-PointClTop(X) holds A is closed iff x0 in A
proof
  let X be non empty set;
  let x0 be Element of X;
  let A be non empty Subset of x0-PointClTop(X);
  A is closed iff Cl A = A by PRE_TOPC:22;
  then A is closed iff A = A \/ {x0} by Th37;
  then A is closed iff {x0} c= A by XBOOLE_1:7,12;
  hence thesis by ZFMISC_1:31;
end;
