On the Decompositions of Intervals and Simple Closed Curves1

The aim of the paper is to show that the only subcontinua of the Jordan curve are arcs, the whole curve, and singletons of its points. Additionally, it has been shown that the only subcontinua of the unit interval I are closed intervals. [9] provide the notation and terminology for this paper. 1. PRELIMINARIES One can check that every simple closed curve is non trivial. Let T be a non empty topological space. One can check that there exists a subset of T which is non empty, compact, and connected. Let us observe that every element of I is real. One can prove the following two propositions: (1) Let X be a non empty set and A, B be non empty subsets of X. If A ⊂ B, then there exists an element p of X such that p ∈ B and A ⊆ B \ {p}. (2) Let X be a non empty set and A be a non empty subset of X. Then A is trivial if and only if there exists an element x of X such that A = {x}. Let T be a non trivial 1-sorted structure. Note that there exists a subset of T which is non trivial. The following proposition is true (3) For every non trivial set X and for every set p there exists an element q of X such that q = p. Let X be a non trivial set. One can verify that there exists a subset of X which is non trivial. We now state a number of propositions: (4) Let T be a non trivial set, X be a non trivial subset of T , and p be a set. Then there exists an element q of T such that q ∈ X and q = p.