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A family of pseudometrics defined on a set gives rise to a uniform structure on that set. Moreover (up to uniform equivalence, anyway) every uniform structure arises this way. Let $A$ and $B$ be families of pseudometrics defined on sets $X$ and $Y$ respectively. There ought to be a way to characterize the uniform continuity of maps $X$ to $Y$ directly in terms of $A$ and $B$ and I thought I knew the correct way to do this but, when I scanned through the relevant sections in Dugundji's Topology to confirm my suspicions, I came across:

"$f: X \to Y$ is uniformly continuous if for each $\beta \in B$ and each $\epsilon > 0$, there exists an $\alpha \in A$ and a $\delta > 0$ such that if \alpha(x,x') < \delta, then \beta(f(x),f(x')) < \epsilon."

This seems stronger than uniform continuity to me. When I tried to work out the right condition I found that one might, given an $\epsilon > 0$ and a $\beta \in B$, only be able to find a $\delta>0$ and finitely many pseudometrics $\alpha_1,\ldots,\alpha_n \in A$ such that (\alpha_i(x,x') < \delta \ \ \forall i) \Rightarrow \beta(f(x),f(x')) < \epsilon.

Thoughts?

Edit: I don't think there's much more one can say in response to this question. If $\mathscr{D}$ is a family of pseudometrics on $X$, one can safely replace $\mathscr{D}$ with the family $\mathscr{D}^+$ consisting of all pseudometrics of the form $(x,y) \mapsto \max_{d \in \mathscr{F}} d(x,y)$ where $\mathscr{F}$ is some finite subset of $\mathscr{D}$ without inducing a finer uniform structure on $X$. As remarked by Theo Buehler in the comments, if we assume, with no loss of generality so far as the uniform structure on $X$ is concerned, that my $A = A^+$ then Dugundji's definition of uniform continuity is equivalent to the clumsier one. In summary, there is a small inaccuracy in Dugundji which can be easily corrected by the addition of a single superscript $+$ to make explicit an assumption that the author (rather harmlessly I might add) probably considered implicit. I would call the matter settled. If anyone disagrees, feel free to say so, but you will do so too late to save my hapless copy of Dugundji - which has already been permanently defaced!

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    @Sam: I don't have a copy of Dugundji's book at the moment, but I seem to recall having the impression that Dugundji was *defining* uniform continuity in terms of the pseudometrics so the "only if" was implied. Or possibly I misquoted the passage? Certainly, stated out of context the way it is above, your point is valid. Good work for noticing!2011-05-02

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This was resolved in the comments