Subspace Topology of a subset
Solution 1:
Let $\tau_{X}$ and $\tau_{Y}$ be the topologies of A as subspace of X and Y.
If $U \in \tau_{X} \Rightarrow \exists B $ open in X such that $U = B \cap A = (B\cap Y) \cap A$ and then $U \in \tau_{Y}$.
If $U \in \tau_{Y} \Rightarrow \exists B $ open in Y such that $U = B \cap A \Rightarrow \exists C $ open in X such that $ B = C \cap Y \Rightarrow U = (C \cap Y)\cap A = C \cap ( Y \cap A) = C \cap A.$
Then $\tau_{X} =\tau_{Y}$.