reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem
  for p being Point of Tunit_circle(2) holds Topen_unit_circle(p), I(01)
  are_homeomorphic
proof
  set D = Sphere(0.TOP-REAL 2,p1);
  let p be Point of Tunit_circle(2);
  set P = Topen_unit_circle(p);
  reconsider p2 = p as Point of TOP-REAL 2 by PRE_TOPC:25;
  D\{p} c= D by XBOOLE_1:36;
  then reconsider A = D\{p} as Subset of Tcircle(0.TOP-REAL 2,1) by Th9;
  P = Tcircle(0.TOP-REAL 2,1) | A by Lm13,Th22,EUCLID:54
    .= (TOP-REAL 2) | (D \ {p2}) by GOBOARD9:2;
  hence thesis by Lm13,BORSUK_4:52,EUCLID:54;
end;
