EDB — 2F7

[2F7]Prerequisites:[0M3]↺↻,[0M5]↺↻,[0M7]↺↻.

Let, more in general, \(I\) be a non-empty index set, and let \((X_ i,\tau _ i)\) be topological spaces, for \(i\in I\); let \({\mathcal B}_ i\) be a base for \(\tau _ i\). (Note that the choice \({\mathcal B}_ i=\tau _ i\) is allowed.)

Let \(X=∏_{i\in I} X_ i\) be the Cartesian product.

We define the product topology \(𝜏\) on \(X\), similarly to [0M3]↺↻, but with a twist.

A base \({\mathcal B}\) for \(\tau \) is the family of all sets of the form \(A=∏_{i\in I} A_ i\) where

and moreover \(A_ i= X_ i\) but for finitely many \(i\).

Show that \({\mathcal B}\) satisfies the requirements in [0KX]↺↻, so it is a base for the topology \(\tau \) that it generates. Show that the product topology does not depend on the choice of the bases \({\mathcal B}_ i\).