## “Completeness” under uniform continuity [closed]

We say that a metric space where every convergent sequence converges to something in the space is called "complete" and has useful properties, but is there a specific name for spaces which fail to be complete, but have at least the condition that every uniformly convergent sequence converges to something in the space?

Like, for instance, the metric space of polynomials over R is not complete as far as I know, but if you restrict yourself to uniformly convergent sequences, then you always get a polynomial. Is there a name for that, or are these types of spaces just not that interesting?

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When you say "uniformly", over what set are you imposing uniformness of the convergence? If you're requiring uniform convergence over the whole real line, this is extremely restrictive, because the only bounded polynomials are constant. – Yemon Choi Nov 12 2009 at 3:33
A polynomial sequence converges uniformly over R if and only if every term except the constant term is eventually constant and the constant term converges. – Qiaochu Yuan Nov 12 2009 at 4:32
I should've said this more clearly: "uniformly convergent" is meaningless in a general metric space. One usually needs a space of functions into a metric space, or more generally a uniform space. – Qiaochu Yuan Nov 12 2009 at 4:41
Also, "the metric space of polynomials over R" is meaningless until you specify what metric you're using, and the uniform metric isn't even defined everywhere. Probably the example you were thinking of is that the uniform limit of continuous functions is continuous. – Qiaochu Yuan Nov 12 2009 at 4:45
"where every convergent sequence converges to something". This is the definition. A complete metric space is one in which every Cauchy sequence converges. The Cauchy condition says something like "ought to converge". Also, as other commenters have observed, the pointwise-convergence topology and the uniform-convergence topoloy are simply different topologies on the space of functions. I don't know of natural metrics that give either, but they are probably metrizable. – Theo Johnson-Freyd Nov 12 2009 at 8:47