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As has already been remarked by Lasse Rempe-Gillen, you need to know what a Cauchy net is, say from a uniform structure. But since you want $\kappa$ to be complete discrete [edit: I had complete’ here once] as a topological space, surely you just want it to be discrete as a uniform space, so let's do that. And then give $\kappa^\kappa$ the product uniformity. (For $\kappa := \omega$, this is the uniform structure that underlies the usual metric on Baire space.)

So the answer is Yes, $\kappa^\kappa$ is complete. This is because every discrete uniform space is complete, and (as Lasse has remarked) any product of complete spaces is complete. Also, note that the underlying topology of the product uniformity is the product topology, so $\kappa^\kappa$ has the topology that you originally wanted.

The cited facts are all part of my general knowledge; I will try to find specific references for them.

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As has already been remarked by Lasse Rempe-Gillen, you need to know what a Cauchy net is, say from a uniform structure. But since you want $\kappa$ to be complete as a topological space, surely you just want it to be discrete as a uniform space, so let's do that. And then give $\kappa^\kappa$ the product uniformity. (For $\kappa := \omega$, this is the uniform structure that underlies the usual metric on Baire space.)

So the answer is Yes, $\kappa^\kappa$ is complete. This is because every discrete uniform space is complete, and (as Lasse has remarked) any product of complete spaces is complete. Also, note that the underlying topology of the product uniformity is the product topology, so $\kappa^\kappa$ has the topology that you originally wanted.

The cited facts are all part of my general knowledge; I will try to find specific references for them.