$C[0,1]=$ space of all continuous complex valued function over $[0,1]$.
Define metric,
$d(f,g)={\int_{0}^{1} \frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}$, for all $f,g\in C[0,1] .$
Let $(C[0,1],\sigma)$ be the topology induced by $d$.
How to prove this topology has only convex open sets $\emptyset$ and $C[0,1]$.