Q:

Is $\left[ 0,1 \right)$ a completely metrizable space? 

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 $\left[ 0,1 \right)$? 

A:

It is possible : Note that $[0,\infty)$ is complete with an induced metric from $\mathbb{R}$ So $[0,1)$ is complete

