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This paper is devoted to study the space A(U) of all analytic functions on an open subset U of RN or CN. It is proved that if U satisfies a weak condition (that will be called 0-property), then every f∈A(U) depends only on a finite number of variables. Then several topologies on A(U) are studied: the compact open topology, the τδ topology (already known in spaces of holomorphic functions) and a new one, defined by the inductive limit of the subspaces of analytic functions which only depend on the first variables.