 Solution

If $A\subseteq X$, the subspace topology on $A$ isIf $\sim$ is an equivalence relation on $X$ and $q:X\to X/\sim$ is the natural projection, the quotient topology declares $V\subseteq X/\sim$ open exactly when $q^{-1}(V)$ is open in $X$. For topological spaces $X$ and $Y$, the product topology on $X\times Y$ has as a basis all sets $U\times V$ with $U$ open in $X$ and $V$ open in $Y$.