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We prove that the set of Segre-degenerate points of a real-analyticsubvariety $X$ in ${\mathbb{C}}^n$ is a closed semianalytic set. It is asubvariety if $X$ is coherent. More precisely, the set of points where the germof the Segre variety is of dimension $k$ or greater is a closed semianalyticset in general, and for a coherent $X$, it is a real-analytic subvariety of$X$. For a hypersurface $X$ in ${\mathbb{C}}^n$, the set of Segre-degeneratepoints, $X_{[n]}$, is a semianalytic set of dimension at most $2n-4$. If $X$ iscoherent, then $X_{[n]}$ is a complex subvariety of (complex) dimension $n-2$.Example hypersurfaces are given showing that $X_{[n]}$ need not be a subvarietyand that it also needs not be complex; $X_{[n]}$ can, for instance, be a realline.

Segre-Degenerate Points Form a Semianalytic Set