
theorem Th4:
  for T being lower non empty TopRelStr for x being Point of T
  holds (uparrow x)` is open & uparrow x is closed
proof
  let T be lower non empty TopRelStr;
  set BB = the set of all (uparrow x)` where x is Element of T;
  let x be Point of T;
  reconsider a = x as Element of T;
  BB is prebasis of T by Def1;
  then
A1: BB c= the topology of T by TOPS_2:64;
A2: (uparrow a)` in BB;
  hence (uparrow x)` in the topology of T by A1;
  thus [#]T \ uparrow x in the topology of T by A2,A1;
end;
