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I am trying to generalize the notion of reorientation class of an oriented matroid to the context of matroids over hyperfields (compare Baker and Bowler, 2016). I have already got some results in this direction. Particularly, given a hyperfield $\mathbb{H}$ and a matroid $M$ I am able to combinatorially and algebraically characterize the "moduli space" of all "reorientation" classes of $\mathbb{H}$-matroids having $M$ as underlying matroid.

It would be nice to extend these statements to the setting of "topological" hyperfields, improving on some results on phased matroids (compare Delucchi and S., 2015). However, I do not know a proper definition of these objects.

Question: Is there any definition of topological hyperfield? In that case, can you kindly provide some reference in the literature?

Thank you very much for any kind of comments or suggestions.

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