I haven't thought about this really hard, so it might not be correct, but you can try letting under the usual Euclidean metric. Then we can represent . In other words, each copy of allows you to write down a digit of a number from . For example,
At this point, it is necessary to check that the product topology for is the same as the Euclidean topology, which I have not done. Maybe you can think about it. If it is true, then Tychonoff's theorem will finish the problem.