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A subset $E$ of $\\Omega$ is said to be a convergence set in $\\Omega$ if there is a series $f(z,t)$ such that $E$ is exactly the set of points $z$ for which $f(z,t)$ converges as a power series in a single variable $t$ in some neighborhood of the origin. A $\\sigma$-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. 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