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Product Topology**

Given two topological spaces \((X,\mathcal{T}_X),\:(Y,\mathcal{T}_Y)\), one defines the product topology on their Cartesian product \(X\times Y\), \(\mathcal{T}_{X\times Y}\), as the coarsest topology on \(X\times Y\) to contain all sets \(U\times V\) where \(U\in\mathcal{T}_X\) and \(V\in\mathcal{T}_Y\), i.e. the topological space with basis \(\{U\times V\subseteq X\times Y|U\in\mathcal{T}_X\land V\in\mathcal{T}_Y\}\). The space \((X\times Y,\mathcal{T}_{X\times Y})\) is the product of the aforementioned spaces in the categorical sense (with projection maps \(\pi_1:X\times Y\to X,(x,y)\mapsto x,\:\:\pi_2:X\times Y\to Y,(x,y)\mapsto y\).