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I'm studying a topology book and it states that metric spaces are not enough to study proximity and convergence, arguing that there is no way to define a metric on the space of all real functions on the interval $[0,1]$ so that the $\{f_n\}$ converges pointwise to $f$ if and only if the distance between $f_n$ and $f$ converges to zero, but the authors don't give any support to their claim.

Can you help me out trying to figure out why? Thank you!

mrtaurho
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Nicolas
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    There is a metric on any set. But there is no metric on the space of all real functions on $[0,1]$ such that $f_n \to f$ in this metric iff $f_n(x) \to f(x)$ for each $x$. I suppose they are referring to this result. – Kavi Rama Murthy Aug 18 '21 at 09:05
  • In addition to Kavi Rama Murthy's comment : https://math.stackexchange.com/questions/1097835/metric-on-function-space-for-pointwise-convergence – TheSilverDoe Aug 18 '21 at 09:06
  • @KaviRamaMurthy You're right, thanks for pointing that out. Do you know where I can find a proof of the result you're referring to? – Nicolas Aug 18 '21 at 09:13
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    There is a proof here: https://math.stackexchange.com/questions/179800/cx-with-the-pointwise-convergence-topology-is-not-metrizable – Kavi Rama Murthy Aug 18 '21 at 09:21
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    $\Bbb R^{[0,1]}$ is the set of functions from $[0,1]$ to $\Bbb R$ and if we give it the unique topology such that convergence (of nets) is just pointwise convergence, this means this set gets the canonical product topology which is not first countable at any point, so very far from metrisable. – Henno Brandsma Aug 18 '21 at 09:34

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