On topological space $ X =(0,1) ^{2}$ with dictionary order topology find the weight of the space $X$. On topological space $ X =(0,1) ^{2}$  with dictionary order topology find the weight of the space $X$.
I am simply stuck don't even know where to start. 

You mean $(0,1)^2$ (from the comments) and the open sets $\{x\} \times (0,1), x \in (0,1)$ (open intervals between $(x,0)$ and $(x,1)$ so they're order-open) are non-empty and pairwise disjoint. So any base has size at least $|(0,1)|=\mathfrak{c}$. The standard base for that space (open intervals) also has that size so $w((0,1)^2) = \mathfrak{c}$.
