The definition of completely metrizable space can refer https://en.wikipedia.org/wiki/Completely_metrizable_space
As we know,with usual metric, $\left[ 0,1 \right]$ is a complete metric space. So $\left[ 0,1 \right]$ with usual topology is a completely metrizable space.
Just as Wikipedia said, $\left( 0,1 \right)$ is a completely metrizable space although its usual metric is not complete.
So, what about $\left[ 0,1 \right)$?
It is possible : Note that $[0,\infty)$ is complete with an induced metric from $\mathbb{R}$ So $[0,1)$ is complete