Suppose \(X_1,\ldots , X_n\) are topological spaces . The Cartesian product \(X_1\times \cdots \times X_n\) endowed with the topology generated by
\begin{equation*} \{U_1\times \cdots \times U_n \mid U_i\subseteq X_i \text{ open }\} \end{equation*}is known as a product space.
Remarks
- We could also consider the topology generated by \(\{B_1\times \cdots B_n \mid B_i\in \mathcal{B}_i\}\) (where \(\mathcal{B}_i\) is a basis of \(X_i\)). Both topologies are indeed identical.
- The product space topology satisfies the characteristic property of product spaces
Examples Link to heading
- \(\mathbb{R}^n\) is the product of \(n\) copies of \(\mathbb{R}\)
- torus