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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.

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Engelking´s "General Topology" is a good reference for all results mentioned in Lasse and Toby´s answers. Also you can find there the notion of Cech-complete space (a generalization of completely metrizable but in a different direction) and the fact that not even $\omega^{\omega_1}$ is Cech-complete. –  Ramiro de la Vega Nov 13 '12 at 23:24
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2 Answers 2

up vote 4 down vote accepted

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.

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@Lasse: So, my question becomes is there a way that $\kappa^\kappa$ be given a uniform structure so that (1) $\kappa^\kappa$ is complete and (2) the topology induced by the uniformity is the product topology? –  Ioannis Souldatos Nov 13 '12 at 22:13
    
As Toby states, the answer is yes. The topology underlying the product uniformity will be the product topology. –  Lasse Rempe-Gillen Nov 13 '12 at 22:32
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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 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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The encyclopedia entry quotes Kelley as one of its sources, and I would guess the statement is in there (though it's a long, long time that I read about uniform spaces in Kelley). I've got the book in my office and can try to remember to check tomorrow, if nobody beats me to it! –  Lasse Rempe-Gillen Nov 13 '12 at 22:34
    
Thank you all for the answers. –  Ioannis Souldatos Nov 14 '12 at 19:24
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