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"?
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.