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Let $\mathcal{C}(R_+)$ be the space of continuous functions $f$ defined on $[0,+\infty)$ with $f(0)=0$. Denote by $\Omega$ the product of $R_+$ and $\mathcal{C}(R_+)$. Now endow $\Omega$ with the following metric: for any $(t,f), (t',f')\in\Omega$ define

$$d\big((t,f),(t',f')\big)=|t-t'|+||f_t-f'_{t'}||$$

where $||\cdot||$ stands for the uniform norm, i.e. $||g||=\sup_{u\in R_+}|g(u)|$ and $g_t\in\mathcal{C}(R_+)$ is defined by $g_t(u)=g(\min(t,u))$ for $u\ge 0$. My question is whether $\Omega$ is separable for the distance $d$? Many thanks for the answer.

WARNING:   The distance function $\ d\ $ (as it is now, as of 2014-09-05) is not a metrics but a pseudo-metrics. See a respective comment underneath.

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    $\begingroup$ That metric looks very unusual to me. Would you mind explaining where it comes from and what you need the separability for? $\endgroup$
    – Joonas Ilmavirta
    Commented Sep 4, 2014 at 21:43
  • $\begingroup$ The question is from the study of stochastic process defined on an enlarged space $R_+\times\mathcal{C}(R_+)$. In order to study the convergence, I would like to find some metris under which this space is Polish. $\endgroup$
    – CodeGolf
    Commented Sep 4, 2014 at 23:03
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    $\begingroup$ As I've mentioned under the @Bjørn's answer, metrics $d$ is only a pseudo-metrics. Indeed, consider functions $\ f(x) := x\ $ and $\ g(x):= 1 - |x-1|.\ $ Then $\ d((1\ f)\ (1\ g)) = 0,\ $ while $\ (1\ f)\ne(1\ g)$. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 5, 2014 at 4:19
  • $\begingroup$ This space originates from the study of path-dependent PDE, which is an interpretation of SDE with random coefficients. $d$ is just a pseudo-metric. $\endgroup$
    – Morris
    Commented Oct 29, 2014 at 23:10

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Sure, consider the set of pairs $(q,r)$ where $q$ is rational and $r$ is a piecewise linear continuous function with rational break-points (and rational values at the breakpoints) with finitely many pieces, and such that the rightmost piece is a constant.

This set is countable and dense.

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    $\begingroup$ Thank you @Bjørn for your theorem, and for correcting my error (I had my dark moments). However, the metrics is not well defined. Certainly different pairs $\ (s\ f)\ $ and $\ (t\ g)\ $ may still have distance $0$, say when $\ s=t\ $ and $f|[0;s] = g|[0;s].\ $ (Just in case one should check the triangle inequality too :-). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 5, 2014 at 4:02

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