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.)?