EDB — 2F9

[2F9]Prerequisites:[2F5]↺↻,[2F7]↺↻,[071]↺↻,[2F7]↺↻.

Consider totally ordered sets \((X_ i,≤_ i)\) (each has at least two elements), and the associated order topologies \(𝜏_ i\).

Let \(I=ℕ\) or \(I=\{ 0,1,\ldots N\} \); let \(X=∏_{i∈ I} X_ i\) be the Cartesian product.

Consider these two topologies.

• We define the product topology \(𝜏\) on \(X\), as explained in [2F7]↺↻.

• We order \(X\) with the lexicographical order \(⪯\), and then we build the order topology \(𝜎\) on \(X\). (See [071]↺↻,[2F7]↺↻)

Is there an inclusion between \(𝜎\) and \(𝜏\)?

If every \(X_ i\) is finite, prove that these two topologies coincide  ¹ .