What does it mean to say that $d$ is a metric on a set $X$ ? What does it mean to say that a subset of $X$ is open with respect to the metric $d$ ? Show that the collection of subsets of $X$ that are open with respect to $d$ satisfies the axioms of a topology.

For $X=C[0,1]$, the set of continuous functions $f:[0,1] \rightarrow \mathbb{R}$, show that the metrics

$$\begin{aligned} &d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| \mathrm{d} x \\ &d_{2}(f, g)=\left[\int_{0}^{1}|f(x)-g(x)|^{2} \mathrm{~d} x\right]^{1 / 2} \end{aligned}$$

give different topologies.