Pregeometry over locally o-minimal structure and dimension

Masato Fujita

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arXiv · 2204.01895

Pregeometry over locally o-minimal structure and dimension

Masato Fujita

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Abstract

We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact. A definable set $X$ is of dimension equal to the rank of $X$ over the set of parameters of a formula defining the set $X$. The structure $\mathcal M$ is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure $\mathcal M$ also coincides with its dimension.

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BibTeX RIS

Masato Fujita. 2022-04-04. Pregeometry over locally o-minimal structure and dimension. https://arxiv.org/abs/2204.01895

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