I am assuming that uniform spaces are Hausdorff (although it probably doesn't matter for this question). It is more-or-less obvious that a uniform space can be embedded in a product of metric space (if d is a semi-metric on the space) form the quotient gotten by identifying pairs of points at d-distance 0 and then map into the product of these quotients), but I would be interested either to know that every uniform space can be embedded as a closed subspace of such a product or an example of one that cannot.

## 1 Answer

You are looking for the notion of Dieudonne complete spaces (which turn out to be exactly the closed subspaces of products of metric spaces). As Todd mentioned in his comment, this notion is closely related with the notion of realcompactness (the two notions coincide if there are no measurable cardinals). A way to find examples is to look for pseudo-compact non-compact spaces (for instance $\omega_1$ with (a uniformity compatible with) the order topology).

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