The Brouwer Fixed Point Theorem for Intervals 1 Toshihiko Watanabe Shinshu

The following three propositions are true: (1) For all real numbers a, b, c, d such that a ≤ c and d ≤ b and c ≤ d holds [c, d] ⊆ [a, b]. (2) For all real numbers a, b, c, d such that a ≤ c and b ≤ d and c ≤ b holds [a, b] ∪ [c, d] = [a, d]. (3) For all real numbers a, b, c, d such that a ≤ c and b ≤ d and c ≤ b holds [a, b] ∩ [c, d] = [c, b]. In the sequel a, b, c, d are real numbers. We now state four propositions: (4) For every subset A of 1 such that A = [a, b] holds A is closed. (5) If a ≤ b, then [a, b]T is a closed subspace of 1 . (6) If a ≤ c and d ≤ b and c ≤ d, then [c, d]T is a closed subspace of [a, b]T. This paper was done under the supervision of Z. Karno while the author was visiting the Institute of Mathematics of Warsaw University in Bia lystok.