$(X_1,\tau_1)$ and $(X_2.\tau_2)$ are topological spaces with $Y_1\subset X_1; Y_2\subset X_2$. Let $X_1\times X_2=X; Y_1\times Y_2=Y $. Prove that the product topology on $Y$ obtained from topologies $\tau_i|Y_i$ is the same as the relativization to $Y$, the product topology on $X$.
In the above question what do we mean by relativization and what to show, I am not getting the meaning of the question. Can anyone help ?
The set $Y$ is a subset of $X$. Therefore, if you consider to product topology on $X$, a natural topology to consider on $Y$ is the relative topology: if $A\subset Y$, then $A$ is open if $A=Y\cap A^\star$, for some open subset $A^\star$ of $X$.
On the other hand, since $Y_1$ and $Y_2$ are subsets of $X_1$ and $X_2$ then $Y_1$ and $Y_2$ get the relativa topology from $X_1$ and $X_2$ respectively. And then you have the product topology on $Y_1\times Y_1$.
The problem is to prove that these two topologies on $Y_1\times Y_2$ are equal.