My question is:
Is every Polish space image of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$?
Where a Polish space is a separable and completely metrizable space and where $\Bbb{N}^\Bbb{N}$ (aka Baire space) is the space of infinite sequences of natural numbers endowed with the product of the discrete topology over $\Bbb{N}$.
I know that every Polish space is image of a continuous bijection with domain a closed subset of $\Bbb{N}^\Bbb{N}$ (hence, in particular, it is a continuous image of $\Bbb{N}^\Bbb{N}$). But what happens if we require the mapping to be closed?
Moreover, in case the answer to the above question is "no", is there a counterexample? Can we characterize the Polish spaces that are images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$?
Hints? Ideas?
Thanks!
