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").

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.