I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy question, but it is not my area and I did not find a reference.

**Definitions**: Let $\kappa$ be an *uncountable* cardinal equipped with the discrete topology and $\kappa^\kappa$ the set of all functions from $\kappa$ to $\kappa$ equipped with the product topology.

A Hausdorff topological space is complete is every Cauchy net converges to a (unique) limit.

**Question**: Is $\kappa^\kappa$ complete?

If so can you provide some reference (paper, textbook etc.)?

Thank you.