Product Topology

In the setting of Sets and Maps: Ordinary Notation, let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ be topological spaces.

(i) The product topology on $X\times Y$ is the set $\mathcal{T}_{X\times Y}$ of all $W\in\mathcal{P}(X\times Y)$ such that for all $x\in X$ and $y\in Y$ with $(x,y)\in W$ there are $U\in\mathcal{T}_{X}$ and $V\in\mathcal{T}_{Y}$ with $x\in U$, $y\in V$ and $U\times V\subseteq W$. It is a topology on $X\times Y$ by The Product Topology is a Topology (i); thus $(X\times Y,\mathcal{T}_{X\times Y})$ is a topological space.

(ii) A subset $W$ of $X\times Y$ is open in $(X\times Y,\mathcal{T}_{X\times Y})$ if and only if for all $x\in X$ and $y\in Y$ with $(x,y)\in W$ there are $U\in\mathcal{T}_{X}$ and $V\in\mathcal{T}_{Y}$ with $x\in U$, $y\in V$ and $U\times V\subseteq W$.