Intermediate Value Theorem and Thickness of Simple Closed Curves

For simplicity, we adopt the following convention: a, b, c, d, r1, r2, r3, r, r4, s1, s2 are real numbers, p, q are points of E 2 T , P is a subset of the carrier of E2 T , and X, Y , Z are non empty topological spaces. Next we state a number of propositions: (1) For all a, b, c holds c ∈ [a, b] iff a ¬ c and c ¬ b. (2) Let f be a continuous mapping from X into Y and g be a continuous mapping from Y into Z. Then g · f is a continuous mapping from X into Z. (3) Let A, B be subsets of the carrier of X. If A is open and B is open and A ∩ B = ∅X , then A and B are separated.