Journal of Formalized Mathematics a Decomposition of Simple Closed Curves and an Order of Their Points

The goal of the article is to introduce an order on a simple closed curve. To do this, we x two points on the curve and devide it to two arcs. We prove that such a decomposition is unique. Other auxiliary theorems about arcs are proven for preparation of the proof of the above. For simplicity, we use the following convention: a, b, c, s, r are real numbers, n is a natural number, p, q are points of E 2 T , and P is a subset of the carrier of E 2 T. We now state a number of propositions: (1) If a = a+b 2 ; then a = b: (2) If r s; then r r+s 2 and r+s 2 s: (3) Let T 1 be a non empty topological space, P be a subset of the carrier of T 1 , A be a subset of T 1 ↾P; and B be a subset of T 1. If B is closed and A = B \ P; then A is closed. (4) Let T 1 , T 2 be non empty topological spaces, P be a non empty subset of the carrier of T 2 , and f be a map from T 1 into T 2 ↾P: Then (i) f is a map from T 1 into T 2 , and (ii) for every map f 2 from T 1 into T 2 such that f 2 = f and f is continuous holds f 2 is continuous. (5) For every real number r and for every subset P of E 2 T such that P = fp; p ranges over points of E 2 T : p 1 rg holds P is closed. (6) For every real number r and for every subset P of E 2 T such that P = fp; p ranges over points of E 2 T : p 1 rg holds P is closed. (7) For every real number r and for every subset P of E 2 T such that P = fp; p ranges over points of E 2 T : p 1 = rg holds P is closed.