Finite-to-one maps into Euclidean manifolds and spaces with disjoint disks properties

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It is shown that every Euclidean manifold M has the following property for any m ⩾ 1 : If f : X → Y is a perfect surjection between finite-dimensional metric spaces, then the mapping space C ( X , M ) with the source limitation topology contains a dense G δ -subset of maps g such that dim B m ( g ) ⩽ m dim f + dim Y − ( m − 1 ) dim M . Here, B m ( g ) = { ( y , z ) ∈ Y × M | | f − 1 ( y ) ∩ g − 1 ( z ) | ⩾ m } . The existence of residual sets of finite-to-one maps into product of manifolds and spaces having disjoint disks properties is also obtained.