
theorem
  for T being non empty TopSpace, p being Point of T holds p is isolated
  iff {p} is open
proof
  let T be non empty TopSpace, p be Point of T;
A1: {p} /\ [#]T = {p} by XBOOLE_1:28;
  hereby
    assume p is isolated;
    then p is_isolated_in [#]T;
    then ex G being open Subset of T st G /\ [#]T = {p} by Th22;
    hence {p} is open;
  end;
  assume {p} is open;
  then p is_isolated_in [#]T by A1,Th22;
  hence thesis;
end;
