Find $\operatorname{Cl }A$ in topological space $\mathbb{R^2}$ with dictionary order topology. Let $ A = \left\{  (x,y) \in \mathbb{R^2} \mid  y= \sin ( \frac{1}{x}) , \  0 < x \leq 1 \right\}$ . Find $\operatorname{Cl} A$ in topological space $\mathbb{R^2}$ with dictionary order topology.
I guess $ \operatorname{Cl} A = A $? 

The plane in the dictionary order is basically $\mathbb{R}$ many disjoint topological copies of vertical stalks that are homeomorphic to $\mathbb{R}$. So the closure can be taken in each “stalk”. Your $A$ intersects each stalk in a singleton or the emptyset, so the closure is just $A$ as you claim.
