The study of Scheepers' conjecture on local properties of function spaces and special subsets of reals

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05352
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kanagawa University
• Principal Investigator: SAKAI MASAMI 神奈川大学, 理学部, 教授 (60215598)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 関数空間 / Scheepers予想 / Menger

Outline of Final Research Achievements

The purpose of this study was to solve Scheepers' conjecture on the relation between a topological property of X and a local property of the function space Cp(X) with the topology of pointwise convergence. Concerning Scheepers' conjecture, we obtained the following results. (1) we gave a characterization of X for Cp(X) to be projective Menger, and gave some implications of topological properties appeared in Scheepers' conjecture, (2) we showed that under some weak local property of X and Y, if Cp(X) and Cp(Y) are linearly homeomorphic and X is Menger, the Y is also Menger, where this is a partial answer to Arhangelskii's problem.

Free Research Field

集合論的位相幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究期間ではScheepers予想の最終解決には至らなかったが、周辺の問題の解決に向けてはいくつかの進捗が得られ、今後の研究の進展に寄与すると思われる。特に、研究成果の概要で述べられた(2)の結果は、実数の部分集合の間ではMenger性は関数空間の間の線形同相で保存されることを示し、今後の研究方向を定めるうえで重要と思われる。