Simple Closed Curves

The notation and terminology used here have been introduced in the following articles: [22], [21], [14], [1], [24], [20], [6], [7], [18], [4], [8], [23], [17], [25], [11], [16], [9], [19], [2], [5], [15], [3], [10], [12], and [13]. We follow the rules: p1, p2, q1, q2 will denote points of E 2 T and P , Q, P1, P2 will denote subsets of E 2 T . The following propositions are true: (1) If p1 6= p2 and p1 ∈ E and p2 ∈ E2, then there exist P1, P2 such that P1 is an arc from p1 to p2 and P2 is an arc from p1 to p2 and E = P1∪P2 and P1 ∩ P2 = {p1, p2}. (2) E is compact. (3) For every map f from (E T ) Q into (E T ) P such that f is a homeomorphism and Q is an arc from q1 to q2 and P 6= ∅ and for all p1, p2 such that p1 = f(q1) and p2 = f(q2) holds P is an arc from p1 to p2. Let us consider P . We say that P is a simple closed curve if and only if: (Def.1) P 6= ∅ and there exists a map f from (E 2 T ) E into (E T) P such that f is a homeomorphism.