reserve Y for TopStruct;
reserve X for non empty TopSpace;

theorem Th30:
  for x being Point of X holds {x} is discrete
proof
  let x be Point of X;
  now
    let D be Subset of X;
A1: now
      assume
A2:   D = {};
      hereby
        take G = {}X;
        thus G is open & {x} /\ G = D by A2;
      end;
    end;
A3: now
      assume
A4:   D = {x};
      hereby
        take G = [#]X;
        thus G is open & {x} /\ G = D by A4,XBOOLE_1:28;
      end;
    end;
    assume D c= {x};
    hence ex G being Subset of X st G is open & {x} /\ G = D by A1,A3,
ZFMISC_1:33;
  end;
  hence thesis;
end;
