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Suppose that denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain in is a -uniform domain if and only if is a -uniform domain, and is a uniform domain if and only if also is a uniform domain, whenever is a closed countable subset of satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. ) between each pair of distinct points in has a lower bound greater than or equal to .

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