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E89

[0M5]Prerequisites:[0M3],[0KX],[0KZ].Let now \(X_ 1,\ldots X_ n\) be topological spaces with topologies \(𝜏_ 1,\ldots 𝜏_ n\) respectively and suppose that \({\mathcal B}_ 1,{\mathcal B}_ 2,\ldots {\mathcal B}_ n\) are bases for these spaces. Let \(X=∏_{i=1}^ nX_ i\) be the Cartesian product, and let

\[ {\mathcal B}=\left\{ ∏_{i=1}^ n A_ i : A_ 1∈{\mathcal B}_ 1,A_ 2∈{\mathcal B}_ 2,\ldots A_ n∈{\mathcal B}_ n\right\} \]
The family of all cartesian products of elements chosen from their respective bases. Show that \({\mathcal B}\) is a base for the product topology. (This exercise generalizes the previous [0M3], taking \({\mathcal B}_ i=𝜏_ i\)).

Solution 1

[0M6]

See also the exercise [0QM] for an application to the case of metric spaces.

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Authors: "Mennucci , Andrea C. G." .
Bibliography
Book index
  • space, topological
  • topological space
  • base, (topology)
  • Cartesian product
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