论向量空间上拓扑格的刚性

Order Pub Date : 2023-11-22 DOI:10.1007/s11083-023-09655-5
Takanobu Aoyama
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引用次数: 0

摘要

拓扑域上向量空间上的向量拓扑是一个(不一定是Hausdorff)拓扑,它的加法和标量乘法是连续的。我们证明了如果两个向量空间的拓扑格间的同构保持了向量拓扑,那么同构是由平移、半线性同构和补映射引起的。因此,如果这种同构存在,则系数场作为拓扑场是同构的,并且这些向量空间具有相同的维数。对于保留Hausdorff向量拓扑的向量拓扑格之间的同构,我们也证明了一个类似的刚性结果。为了得到这些结果,我们在矢量拓扑晶格和子空间晶格之间构造了伽罗瓦连接,并使用了仿射几何和射影几何的基本定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the Rigidity of Lattices of Topologies on Vector Spaces

A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and the scalar multiplication are continuous. We prove that, if an isomorphism between the lattices of topologies of two vector spaces preserves vector topologies, then the isomorphism is induced by a translation, a semilinear isomorphism and the complement map. As a consequence, if such an isomorphism exists, the coefficient fields are isomorphic as topological fields and these vector spaces have the same dimension. We also prove a similar rigidity result for an isomorphism between the lattice of vector topologies which preserves Hausdorff vector topologies. To obtain these results, we construct a Galois connection between a lattice of vector topologies and a lattice of subspaces and use the fundamental theorems of affine and projective geometries.

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