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!



Images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$
This question is answered positively in [On closed images of the space of irrationals](https://www.semanticscholar.org/paper/On-closed-images-of-the-space-of-irrationals-Engelking/726a02a1075858b78aa322f1dfb504e858d39cca), by Engelking.


