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Consider a uniform space $X$ given by a gauge $\mathcal D_X$ (this is supposed to be an ideal w.r.t. $\leq$). Since a topological space is uniformable if and only, if it is completely regular, I'm interested in a "half of a metrization theorem":

Using the language of gauges: What is a necessary and sufficient for $X$ to be pseudometrizable?

Strictly, speaking I mean: "When is $X$ homeomorphic to a pseudometric space?"

But related questions are: "When is $X$ isomorphic to a pseudometric space considered as a uniform space?" "When is $\mathcal D_X$ generated by a single pseudometric?". I'm not sure how they relate to each other (besides the obvious implications in one direction).

References to such a theorem (and a proof) would suffice as well.

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    probably a countable generating set (base) is necessary and suficient in gauge terms, as it is for uniformities (this is why first countable topological groups are metrisable etc.) – 2017-01-28
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    @HennoBrandsma I see. I have no idea how I go about proving this, yet. – 2017-01-28
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    It's probably well-known and in he books already. Why reprove it.. – 2017-01-28
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    @HennoBrandsma I don't know where I can find a proof in terms of gauges. (For now) I would like to avoid the terminology associated with uniformities ("entourage" etc.). – 2017-01-28
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    For a countable base $d_n$ of pseudometrics in the gauge, $\sum_n \frac{1}{2^n} \min(d_n, 1)$ is a single pseudo metric that gives the same topology , I think. – 2017-01-28
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    @HennoBrandsma Thank you. This is indeed a sufficient condition (I thought you meant "base" somehow in the sense of topology). I suppose the converse is where the real work begins. – 2017-01-28
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    No, base in the filter sense. The necessity is clear, a pseudo metric space already has a base of 1 gauge! – 2017-01-28
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    @HennoBrandsma Perhaps I'm missing something: If $(X,\mathcal D_X)$ is homeomorphic to a pseudometricspace, then how do I know that $\mathcal D_X$ has a countable base? – 2017-01-28

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Quoting

A Hausdorff uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms, it is metrizable.A Hausdorff uniform space[clarification needed] is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms, it is metrizable.

from wikipedia