Geodesic
On~the countable definability of sets
Matematičeskie zametki, Tome 59 (1996) no. 3, pp. 382-395

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Let $Q$ be a connected set in $\mathbb C^p$ . Denote by $D[Q]$ the set of all domains containing $Q$, and let $W(Q)$ be the set of all convex domains from $D[Q]$. We present tests for classes $D[Q]$ and $W(Q)$ (in the case when $Q$ is convex for the last one) to have a countable basis. The results are expressed in terms of properties of the boundary $\operatorname{Fr}Q$ of the set $Q$. 
@article{MZM_1996_59_3_a6,
     author = {Yu. F. Korobeinik},
     title = {On~the countable definability of sets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {382--395},
     publisher = {mathdoc},
     volume = {59},
     number = {3},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_3_a6/}
}
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Yu. F. Korobeinik. On~the countable definability of sets. Matematičeskie zametki, Tome 59 (1996) no. 3, pp. 382-395. http://geodesic.mathdoc.fr/item/MZM_1996_59_3_a6/