The set of matrices with a repeated eigenvalue is defined by an algebraic equation ${\rm disc}(M)=0$. This is the discriminant in the eigenvalues $$\prod_{i $\prod_{i\lt j}(\lambda_j-\lambda_i)^2,$$ which is a polynomial in the entries of $M$. Because this polynomial is non-trivial, your set is a non-trivial algebraic variety. In particular, it has zero measure and is closed.
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The set of matrices with a repeated eigenvalue is defined by an algebraic equation ${\rm disc}(M)=0$. This is the discriminant in the eigenvalues $$\prod_{i |
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