EDB — 0M3

[0M3]Prerequisites:[0J1]↺↻,[0KX]↺↻,[0KZ]↺↻. Let now \(X_ 1,\ldots X_ n\) be topological spaces with topologies, respectively, \(𝜏_ 1,\ldots 𝜏_ n\); let \(X=∏_{i=1}^ nX_ i\) 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{align*} 𝜏=\Big\{ ⋃_{j∈ J} ∏_{i=1}^ n A_{i,j} : A_{1,j}∈𝜏_ 1,\ldots A_{n,j}∈𝜏_ n∀ j∈ J , J~ \\ \text{arbitrarily chosen sets of indexes} \Big\} ~ ~ . \end{align*}

• \(𝜏\) is the smallest topology that contains Cartesian products of open sets.