Theorem 4.19 in Kechris' Classical Descriptive Set Theory says that the space of continuous functions from a compact metric space to a Polish space is Polish. It is therefore obvious that the space of continuous functions from a compact Polish space to a Polish space is Polish.


The space of continuous function is equipped with a product metric, for example, the Chebyshev metric.



> 
> My question is, can we drop the "compactness"?
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What anomaly would arise?


If we drop the "continuity", then some problems would arise for cardinality. However, with continuity, we don't have this problem.



Is the space of continuous functions from Polish space to Polish space Polish?
As already answered here ([$X$ is Polish and $N$ is countable. Is $N^X$ Polish?](https://mathoverflow.net/questions/351160/x-is-polish-and-n-is-countable-is-nx-polish/351182#351182)):


Let us work in a convenient category of topological spaces, ie assume that we have function spaces and sufficiently many nice spaces around. My favourite such category is the category of $\mathrm{QCB}\_0$-spaces, which are $T\_0$ quotients of countably-based spaces [1].


Inside the category of $\mathrm{QCB}\_0$-spaces, Schröder has obtained a characterization of the Hausdorff spaces $\mathbf{X}$ such that $\mathbb{R}^\mathbf{X}$ is Polish [2]. These spaces are now called coPolish spaces. A Polish space is coPolish iff it is locally compact. So we can weaken compactness to local compactness, but not further.


A space such as $2^{(\mathbb{N}^\mathbb{N})}$ even fails to be countably based.


[1] <https://www.sciencedirect.com/science/article/pii/S0304397501001098>


[2] <https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.200310111>


