EDB — 0M3

[0M3]Prerequisites:[0J1]↺↻,[0KX]↺↻,[0KZ]↺↻. Let now $X_1,\dots X_n$ be topological spaces with topologies, respectively, ${𝜏}_1,\dots {𝜏}_n$; let $X={\prod }_{i=1}^n{X}_i$ be the Cartesian product. We apply the above results to define the product topology $𝜏$: this can be described in two equivalent ways.

• Union of all Cartesian products of open sets  ¹

  $$\begin{array}{r}𝜏=\{{\bigcup }_{j\in J}{\prod }_{i=1}^n{A}_{i,j}:A_{1,j}\in {𝜏}_1,\dots A_{n,j}\in {𝜏}_n\forall j\in J,J\\ \text{arbitrarily chosen sets of indexes}\}.\end{array}$$

• $𝜏$ is the smallest topology that contains Cartesian products of open sets.