prove that every non-empty open set contains an open sphere disjoint from $A$. 
If $A$ is nowhere dense in $(X,d)$ then prove that every non-empty open set contains an open sphere disjoint from $A$.

Suppose that $A$ is n.w.d. in $X$ , and every non-empty open set say , $B$ contains all open sphere $S_r(x),x\in X,r>0$ such that $S_r(x) \cap A\not= \emptyset.$ Then where it contradicts ?

Let B be a non-empty open set. Suppose every open sphere inside B intersects A. Then every point of B belongs to the closure of A. But this contradicts the definition of a nowhere dense set.
