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Let f be a continuous map of a complete separable metric space E onto theirrationals. We show that if a complete separable metric space M containsisometric copies of every closed relatively discrete set in E, then M containsalso an isometric copy of some fiber of f. We shall show also that if allfibers of f have positive dimension, then the collection of closedzero-dimensional sets in E is non-analytic in the Wijsman hyperspace of E.

Isometric embeddings and continuous maps onto the irrationals