In the [nlab entry on uniform spaces](https://ncatlab.org/nlab/show/uniform+space#examples_2) they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped with the uniform structure generated by:
$$
\bigg\{\big\{ (f,g) \,|\, (f(x),g(x))\in U \;\forall x\in X \big\}, U\in \mathfrak{U}\bigg\}\;.
$$


Is then this the same as the uniform structure on the $X$-fold Cartesian product? If not, does it hold for $X$ "small" enough, e.g. finite?



Cartesian powers of uniform spaces
E.g. $C(\mathbb{R} ,[0,1])$ in this function space uniformity is metrisable in this uniformity, using the $\sup$-metric $d(f,g) = \sup \{|f(x) - g(x)|: x \in \mathbb{R}\}$. While the product $\mathbb{R}^{[0,1]}$ is very non-metrisable (not even first countable) in the product topology. These are quite different beasts..
For finite $X$ they amount to the same thing.


