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過去の記録 ～06/27｜次回の予定｜今後の予定 06/28～
| 担当者 | 酒井拓史、長谷川立 |
|---|
2026年06月18日(木)
15:30-17:30 数理科学研究科棟(駒場) 122号室
Paul Larson 氏 (Miami University)
Discontinuous homomorphisms without Hamel bases
Paul Larson 氏 (Miami University)
Discontinuous homomorphisms without Hamel bases
[ 講演概要 ]
A Hamel basis for the real line is a basis for the line over the scalar field of rational numbers. The Axiom of Choice implies that Hamel bases exist. It is a classical fact that every measurable homomorphism from the additive group on the real line to itself is continuous, and therefore is given by multiplication by some real number. However, permutations of Hamel bases naturally give rise to discontinuous homomorphisms. In this talk we will show that this implication cannot be reversed, by forcing to produce a model of ZF in which there exists a discontinuous homomorphism but there is no Hamel basis. This is joint work with Saharon Shelah.
A Hamel basis for the real line is a basis for the line over the scalar field of rational numbers. The Axiom of Choice implies that Hamel bases exist. It is a classical fact that every measurable homomorphism from the additive group on the real line to itself is continuous, and therefore is given by multiplication by some real number. However, permutations of Hamel bases naturally give rise to discontinuous homomorphisms. In this talk we will show that this implication cannot be reversed, by forcing to produce a model of ZF in which there exists a discontinuous homomorphism but there is no Hamel basis. This is joint work with Saharon Shelah.
