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Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form

$$d(f,g)\rightarrow 0\Longrightarrow f(1)\rightarrow g(1)$$

$$M=\{f\in C^+(I): f(0)=0, f(1)\leq 1\}$$

is compact with respect to this distance? Thanks a lot for your help!

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  • $\begingroup$ Which do you mean by "increasing": strictly increasing or nondecreasing? $\endgroup$
    – Robert Israel
    Commented Nov 28, 2013 at 15:52
  • $\begingroup$ I prefer "nondecreasing" and I wonder under which condition "bounded and closed" sets are compact. $\endgroup$
    – CodeGolf
    Commented Nov 28, 2013 at 20:47
  • $\begingroup$ I think that it is nearly impossible, because for a general given distance $d$, I think that we find a associated norm and then $C^+(I)$ is thus a normed space. $\endgroup$
    – CodeGolf
    Commented Nov 28, 2013 at 20:52
  • $\begingroup$ By Riesz's theorem, the closed unit ball is compact if and only if the space $C^+(I)$ is finite dimensional. $\endgroup$
    – CodeGolf
    Commented Nov 28, 2013 at 20:54

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