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Seminar information archive ～06/27｜Next seminar｜Future seminars 06/28～
| Organizer(s) | SAKAI Hiroshi, HASEGAWA Ryu |
|---|
2026/06/18
15:30-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Paul Larson (Miami University)
Discontinuous homomorphisms without Hamel bases
Paul Larson (Miami University)
Discontinuous homomorphisms without Hamel bases
[ Abstract ]
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.
