EDB — 0QM

[0QM]

Let’s review the exercise [0PT]↺↻.

Having fixed \((X_ 1,d_ 1),\ldots ,(X_ n,d_ n)\) metric spaces, let \(X=X_ 1× X_ 1× \cdots × X_ n\).

Let \(d\) be the distance

This is the same \(d\) defined as in eqn. [(9.26)]↺↻ inside [0PT]↺↻, setting \(𝜑(x)=\max _{i=1\ldots n}|x_ i|\). We indicate with \(B^ d(x,r)\) the ball in \((X,d)\) of center \(x∈ X\) and radius \(r{\gt}0\).

We want to show that \(d\) induces the product topology on \(X\), using the results seen in Sec. [2B5]↺↻.

Taken \(t∈ X_ i, r{\gt}0\) we indicate with \(B^{d_ i}(t,r)\) the ball in metric space \((X_ i,d_ i)\). Let \({\mathcal B}_ i\) be the family of all balls in \((X_ i,d_ i)\).

Let \(\mathcal B\) be defined as

This is the same \({\mathcal B}\) defined in [0M5]↺↻.

Show that every ball \(B^ d(x,r)\) in \((X,d)\) is the Cartesian product of balls \(B^{d_ i}(x_ i,r)\) in \((X_ i,d_ i)\). So let \(\mathcal P\) be the family of balls \(B^ d(x,r)\) in \((X,d)\).

From [0QJ]↺↻ we know that \(\mathcal P\) is a base for the standard topology in the metric space \((X,d)\).

Use [0M7]↺↻ to show that \(\mathcal P\) and \({\mathcal B}\) generate the same topology \(𝜏\).

Use [0M5]↺↻ to prove that \(𝜏\) is the product topology.

We conclude that the distance \(d\) generates the product topology.