Geodesic
Models of set theory in which the separation theorem fails
Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1181-1219

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We use a finite-support product of Jensen-minimal forcings to define a model of set theory in which the separation theorem fails for the projective classes $\mathbf{\Sigma}^1_n$ and $\mathbf{\Pi}^1_n$, for a given $n\geqslant3$.
Keywords: separability, models, Jensen forcing, iteration. 
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     title = {Models of set theory in which the separation theorem fails},
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V. G. Kanovei; V. A. Lyubetsky. Models of set theory in which the separation theorem fails. Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1181-1219. http://geodesic.mathdoc.fr/item/IM2_2021_85_6_a5/