Summary
We examine how in any o-minimal expansion of a dense linear order, fiberwise open implies pecewise open for sets definable with parameters, and fiberwise continuous implies piecewise continuous for functions definable with parameters.
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Speissegger, P. Fiberwise properties of definable sets and functions in o-minimal structures. Manuscripta Math 86, 283–291 (1995). https://doi.org/10.1007/BF02567995
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DOI: https://doi.org/10.1007/BF02567995