On the Decompositions of Intervals and Simple Closed Curves

Let us note 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 the carrier of I is real. Next we state 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}.