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.



Is the generalized Baire space complete?
I think the notion of completeness doesn't make sense for topological spaces; you need at least a uniform structure. 


According to the encyclopedia of mathematics, the product of complete uniform spaces is complete:


<http://www.encyclopediaofmath.org/index.php/Complete_uniform_space>


(I found this by googling "product of complete uniform spaces").


**EDIT.** I just checked my copy of Kelley's "General Topology". Chapter 6 deals with uniform spaces. In this chapter, Theorem 10 states that the topology of the product uniformity is the product topology, and Theorem 25 states that the product of uniform spaces is complete if and only if each coordinate space is complete. This provides the reference you asked for.


