Let \(R\) be a ring and \((M_\alpha)_{\alpha\in A}\) a family of left \(R\)-modules. The direct product of the family is the set \[ \prod_{\alpha\in A} M_\alpha \coloneqq \ts{ (m_\alpha)_{\alpha\in A} \st m_\alpha \in M_\alpha \text{ for all } \alpha\in A } \] with componentwise addition and scalar multiplication, together with the projections \(\pi_\beta\colon\prod_{\alpha\in A} M_\alpha\to M_\beta\), \((m_\alpha)_{\alpha\in A}\mapsto m_\beta\), for \(\beta\in A\).
For every left \(R\)-module \(N\) and every family of \(R\)-linear maps \(f_\beta\colon N\to M_\beta\), \(\beta\in A\), there is a unique \(R\)-linear map \(f\colon N\to \prod_{\alpha\in A} M_\alpha\) with \(\pi_\beta\circ f = f_\beta\) for all \(\beta\in A\), namely \(f(n) = (f_\alpha(n))_{\alpha\in A}\). Thus \(\prod_{\alpha\in A} M_\alpha\) is the categorical product of the family in the category of left \(R\)-modules.
In \(\Top\), the categorical product of a family of spaces \((X_\alpha)_{\alpha\in A}\) is the set \(\prod_{\alpha\in A} X_\alpha\) with the product topology, the coarsest topology for which every projection \(\pi_\beta\) is continuous.
[@DF04]; [@Hat02].