EDB — 0MM

[0MM] ¹ Prerequisites:[2F7]↺↻,[0KK]↺↻.Difficulty:*.Let \(Ω\) be a non-empty set; let’s consider \(X={\mathbb {R}}^Ω\).

1. Let

   \[ U^ f_{E,𝜌}=\{ g∈ X, ∀ x∈ E, |f(x)-g(x)|{\lt}𝜌\} \]

   where \(f∈X\), \(𝜌{\gt}0\) and \(E⊂ Ω\) is finite. Show that the family of these \(U^ f_{E,𝜌}\) satisfies the requirements of [0KZ]↺↻, and is therefore a base for a topology \(𝜏\) (Hint: use [2F7]↺↻). This topology is the product topology of topologies of \(ℝ\).

   In particular for each \(f\in X\) the sets \(U^ f_{E,𝜌}\) are a fundamental system of neighborhoods.

2. Check that the topology is \(T_ 2\).

3. Note that \(X\) is a vector space, and show that the “sum” operation is continuous, as an operation \(X× X→ X\); to this end, show that if \(f,g\in X,h=f+g\), for every neighborhood \(V_ h\) of \(h\) there are neighborhoods \(V_ f,V_ g\) of \(f,g\) such that \(V_ f+V_ g\subseteq V_ h\).

4. Given \(B_ i⊂ℝ\) open and non-empty, one for each \(i∈Ω\), show that \(∏_ i B_ i\) is open if and only if \(B_ i=ℝ\) except at most finitely many \(i\).