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Let $n$ be an integer, and let $D$ be the set of diagonalizable matrices with complex coefficients. Let $S$ be the space of cardinal $n$ multisubsets of $\mathbb{C}$, that is, it is the quotient space of the set of $\mathbb{C}^n$ by all the permutations. Is the map sending a diagonalizable matrix to its spectrum, as a multiset (that is, counting multiplicities), continuous?

I feel this should be true, since I think the only obstruction to continuity of eigenvalues is the fact that when eigenvalues cross, we cannot choose which one to follow. However, I have not been able to prove it.

Plop
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    Good question! I think this follows from this (clearly the characteristic polynomial depends continuously on the matrix), and in fact I reckon your question is close to being equivalent to that question. It's not totally easy to prove! – Izaak van Dongen Aug 22 '24 at 17:37
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    Thanks a lot, it does indeed answer my question :) – Plop Aug 22 '24 at 18:01

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