# Baire's Category Theorem and Some Spaces Generated from Real Normed Space1

Noboru Endou 1 , Yasunari Shidama 2 , and Katsumasa Okamura 2
• 1 Gifu National College of Technology, Gifu, Japan
• 2 Shinshu University, Nagano, Japan

## Baire's Category Theorem and Some Spaces Generated from Real Normed Space1

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 equivalence 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.

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