reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem Th15:
  X is discrete iff for A being Subset of X holds A is open
proof
  thus X is discrete implies for A being Subset of X holds A is open
  by PRE_TOPC:def 2;
  assume for A being Subset of X holds A is open;
  then for V being object
  holds V in the topology of X iff V in bool the carrier of X
  by PRE_TOPC:def 2;
  then the topology of X = bool the carrier of X by TARSKI:2;
  hence thesis;
end;
