if $\displaystyle X $ is a topological space, is $\displaystyle X \times [0,1]$ a topological space, where $\displaystyle [0,1]$ is the unit interval?

Would $\displaystyle X \times [0,\frac{1}{2}] \subseteq X \times [0,1]$ be closed or open or both?

I am interested because im trying to prove that homotopy is an equivalence relation (Transitivity)

Thanks for any help