Baire’s Category Theorem and Some Spaces Generated from Real Normed Space 1.pdf

FORMALIZED MATHEMATICS Volume 14, Number 4, Pages 213–219 University of Bialystok, 2006 Baire’s Category Theorem and Some Spaces Generated from Real Normed Space1 Noboru Endou Yasunari Shidama Gifu National College of Technology Shinshu University Gifu, Japan Nagano, Japan Katsumasa Okamura Shinshu University Nagano, Japan Summary. As application of complete metric space, we proved a Baire’s category theorem. Then we defined some spaces generated from real normed space and discussed each of them. In the second section, we showed the equiv alence of convergence and the continuity of a function. In other sections, we showed some topological properties of two spaces, which are topological space and linear topological space generated from real normed space. MML identifier: NORMSP 2, version: 7.8.03 4.75.958 The papers [23], [7], [26], [4], [1], [21], [15], [27], [6], [5], [17], [19], [20], [24], [22], [2], [25], [9], [10], [13], [16], [12], [11], [3], [18], [8], and [14] provide the notation and terminology for this paper. 1. Baire’s Category Theorem The following proposition is true (1) Let X be a non empty metric space and Y be a sequence of subsets of X . Suppose X is complete and rng Y = the carrier of X and for every element n of N holds Y (n)c ∈ the open set family of X . Then there exists an element n0 of N and there exists a real number r and there exists a point x of X such that 0 r and Ball(x ,r) ⊆ Y (n ). 0 0 0 1This work has been partially supported by the M