$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.


$X^N$ is the collection of all mappings from $N$ to $X$. It is equipped with product topology.



> 
> 1. Is $N^X$ a Polish space?
> 2. Is $X^N$ a Polish space?
> 3. Is $X^X$ a Polish space?
> 
> 
> 


It is obvious that $(2)$ is a Polish space, (3) is not in most cases. I guess $(1)$ is also Polish because $X^N$ and $N^X$ usually have the same structure?



$X$ is Polish and $N$ is countable. Is $N^X$ Polish?
If you really mean these spaces $A^B$ to consist of *all* functions $B\to A$, then (1) and (3) are too big to be Polish. They have cardinality $2^{\mathfrak c}$ where $\mathfrak c$ is the cardinal of the continuum. Separable metric spaces have cardinality at most $\mathfrak c$.


On the other hand, (2) is, like any product of countably many Polish spaces, well known to be Polish.


