Universal finite-to-one map and universal countable dimensional spaces

There is a closed finite-to-one map t́f of a zero-dimensional, separable, metric absolute Gδσ-set X onto a space Y such that for any closed, finite-to-one map f′:X′ → Y′ of separable, metric spaces, with dim X′ ⩽ 0, there exist embeddings i : X′ → X and j : Y′ → Y such that fi = jf. In particular, the space Y is universal for all separable metric spaces which are countable dimensional. We also show that finite-to-one maps produce naturally cell-like maps. Finally, using the method of absorbers we prove a topological characterization of the space σ × N>, where σ is Smirnovʹs universal strongly countable dimensional space and N is Nagataʹs universal countable dimensional space.