I'm working on the second part of following problem (show C(R) finer than subspace topology):
Define $C(\mathbb{R}) \subseteq \mathbb{R}^{\mathbb{R}}$ as the space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Define the metric $\bar{d}: C(\mathbb{R}) \times C(\mathbb{R}) \rightarrow[0, \infty)$ by
$\bar{d}(f, g):=\min \{1, \sup _{x \in \mathbb{R}}|f(x)-g(x)|\}$
Show that $\bar{d}$ is a metric that induces a topology on $C(\mathbb{R})$ that is finer than the subspace topology.
So I know a basis element of topology induced by d is an epsilon ball around some function $f \in C(\mathbb{R})$. I don't really know how to compare this with the intersection of an open ball centered at $f \in \mathbb{R}^\mathbb{R}$ with $C(\mathbb{R})$. We can write a basis element of $\mathbb{R}^\mathbb{R}$ as $\prod_{\alpha}B_\alpha$ where $B_\alpha = X_\alpha$ for all but finitely many $\alpha$. Does it make sense to take $B_\beta \subseteq B_\alpha$ for each $\alpha$ then say $\prod_{\beta}B_\beta \subseteq \prod_{\alpha}B_\alpha \cap C(R)$ for $\prod_{\beta}B_\beta \in C(R)$?
My concern is the existence of these $B_\beta$'s and what they represent (and I'm new to function spaces). I think $B_\beta$ would represent a neighborhood of $f(x_\beta)$ for $x_\beta$ in $\mathbb{R}$?
Sorry if this is unclear and any clarification/help is greatly appreciated.
The sets of the form $C(\mathbb{R})\cap \{g\in \mathbb{R}^\mathbb{R} : g(x_i)\in U_i, i = 1, 2, ..., n\}$ for $x_i\in \mathbb{R}$ and $U_i\subseteq\mathbb{R}$ open, form a basis of $C(\mathbb{R})$ in the subspace topology. If $f\in C(\mathbb{R})$ is an element of such set, by taking $0 < r_i < 1$ such that $(f(x_i)-r_i, f(x_i)+r_i)\subseteq U_i$, and then further taking $r = \min_{i=1, ..., n} r_i$, we can ensure that if $\overline{d}(f, g) < r$, then $g(x_i)\in U_i$ for $i = 1, ..., n$, that is $B(f, r)\subseteq C(\mathbb{R})\cap \{g\in \mathbb{R}^\mathbb{R} : g(x_i)\in U_i, i = 1, 2, ..., n\}$.