EDB — 0PT

[0PT] Prerequisites:[0M3]↺↻, [0PS]↺↻, [107]↺↻, [10F]↺↻,[10J]↺↻.

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

Let \(𝜑\) be one of the norms defined in eqn. [2CK]↺↻ in Sec. [2CK]↺↻. Two possible examples are \(𝜑(x)=|x_ 1|+\cdots + |x_ n|\) or \(𝜑(x)=\max _{i=1\ldots n} |x_ i|\).

Finally, let’s define for \(x,y∈ X\)

Show that \(d\) is a distance; show that the topology in \((X,d)\) coincides with the product topology (see [0M3]↺↻).

Note that this approach generalizes the definition of the Euclidean distance between points in \(ℝ^ n\) (taking \(X_ i=ℝ\) and \(𝜑(z)=\sqrt{∑_ i |z_ i|^ 2}\)). We deduce that the topology of \(ℝ^ n\) is the product of the topologies of \(ℝ\).

See also the exercise [0QM]↺↻, which reformulates the above using the concept of bases of topologies.

[ [0PV]↺↻] [ [0PW]↺↻]