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.

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.

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

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

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.

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.