Is there a reasonably well-behaved topological space $X$ (ideally Polish), a set $\kappa$, and a continuous function $g: X^\kappa\to\mathbb{R}$ that depends on uncountable many coordinates?


If $X$ is a compact Hausdorff space, the answer is known to be no. To see this, note that the family of all continuous functions depending on only finitely many coordinates satisfies the conditions for the Stone-Weierstrass theorem and is, therefore, uniformly dense. The argument can be found in textbooks.



Can a continuous real-valued function on a large product space depend on uncountably many coordinates?
Bockstein's theorem


*Bockstein, M.*, [**Un théorème de séparabilité pour les produits topologiques**](http://dx.doi.org/10.4064/fm-35-1-242-246), Fundam. Math. 35, 242-246 (1948). [ZBL0032.19103](https://zbmath.org/?q=an:0032.19103).


This is the case of a product $\prod\_{t \in T} X\_t$ where all factors are second-countable. I that case any continuous function
$\prod\_{t \in T} X\_t \to \mathbb R$ depends on countably many coordinates.


PLUG... See Theorem 2.1 in


*Edgar, G. A.*, [**Measurability in a Banach space**](http://dx.doi.org/10.1512/iumj.1977.26.26053), Indiana Univ. Math. J. 26, 663-677 (1977). [ZBL0361.46017](https://zbmath.org/?q=an:0361.46017).


where the special case $X = \mathbb R$ is done. That is, a continuous function $\mathbb R^T \to \mathbb R$ depends on only countably many coordinates.


