Inspired by this queston/answer I thought it would natural to conclude that the Space of Riemann integrable functions with the metric of uniform convergence do form a complete metric space,but as I couldn't prove it specially because completeness is not invariant by homeomorphism,is my intuition right?
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1The other answer is your answer. If $x_n$ is Cauchy with the uniform metric, then it has a limit $x$. Since $x_n \to x$ uniformly, we see that $x$ is integrable. What has invariance got to do with it? – copper.hat Dec 13 '16 at 14:27
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I just wanted to mention that if completeness was invariant i could prove by showing such space is homeomorphic to another complete space,which is not the case,in short,Thank you for enlightening me. – AHandsomeAlien Dec 13 '16 at 14:30