Give \(\mathbb{R}^I\) the uniform metric, where \(I = [0, 1]\). Let \(\mathcal{C}(I, \mathbb{R})\) be the subspace consisting of continuous functions. Show that \(\mathcal{C}(I, \mathbb{R})\) has a countable dense subset, and therefore a countable basis. [Hint: Consider those continuous functions whose graphs consist of finitely many line segments with rational end points.]
Solution
Goal: Prove that the metric space \(\mathcal{C}([0, 1], \mathbb{R})\) equipped with the uniform metric is separable and second-countable.
1. Definition of the countable set \(\mathcal{D}\) of rational polygonal functions:Proof:
1.1. Let \(\mathcal{D}\) be the set of all continuous piecewise linear functions \(p: [0, 1] \to \mathbb{R}\) for which:- There exists a finite partition \(0 = t_0 < t_1 < \dots < t_n = 1\) with \(t_i \in \mathbb{Q} \cap [0, 1]\).- The values at the partition nodes are rational: \(p(t_i) \in \mathbb{Q}\) for all \(i \in \{0, \dots, n\}\).- On each subinterval \([t_{i-1}, t_i]\), \(p\) is affine: \(p(t) = p(t_{i-1}) + \frac{p(t_i) - p(t_{i-1})}{t_i - t_{i-1}} (t - t_{i-1})\).
1.2. For each fixed \(n\), the set of such functions is parameterized by \((t_1, \dots, t_{n-1}, y_0, \dots, y_n) \in \mathbb{Q}^{2n}\), which is countable.
1.3. Since \(\mathcal{D} = \bigcup_{n=1}^\infty \mathcal{D}_n\) is a countable union of countable sets, \(\mathcal{D}\) is countable.
2. Density of \(\mathcal{D}\) in \(\mathcal{C}([0, 1], \mathbb{R})\) under the uniform metric:For every \(f \in \mathcal{C}([0, 1], \mathbb{R})\) and every \(\varepsilon > 0\), there exists \(p \in \mathcal{D}\) such that \(\sup_{x \in [0, 1]} |f(x) - p(x)| < \varepsilon\).Proof:
2.1. Since \(I = [0, 1]\) is compact, \(f\) is uniformly continuous on \(I\).
2.2. Choose \(\delta > 0\) such that \(|x - y| < \delta \implies |f(x) - f(y)| < \frac{\varepsilon}{4}\).
2.3. Choose an integer \(n > 1/\delta\) and consider the uniform partition nodes \(x_i = \frac{i}{n} \in \mathbb{Q}\) for \(i \in \{0, \dots, n\}\).
2.4. By density of \(\mathbb{Q}\) in \(\mathbb{R}\), for each \(i \in \{0, \dots, n\}\), choose \(q_i \in \mathbb{Q}\) such that \(|f(x_i) - q_i| < \frac{\varepsilon}{4}\).
2.5. Let \(p \in \mathcal{D}\) be the piecewise linear function connecting \((x_0, q_0), (x_1, q_1), \dots, (x_n, q_n)\).
2.6. For any \(x \in [0, 1]\), choose \(i\) such that \(x \in [x_{i-1}, x_i]\). Write \(x = (1 - \lambda) x_{i-1} + \lambda x_i\) with \(\lambda \in [0, 1]\).
2.7. Then \(p(x) = (1 - \lambda) q_{i-1} + \lambda q_i\), and:\[|p(x) - f(x)| \le (1 - \lambda) |q_{i-1} - f(x)| + \lambda |q_i - f(x)|.\]
2.8. Using the triangle inequality:\[|q_{i-1} - f(x)| \le |q_{i-1} - f(x_{i-1})| + |f(x_{i-1}) - f(x)| < \frac{\varepsilon}{4} + \frac{\varepsilon}{4} = \frac{\varepsilon}{2},\]\[|q_i - f(x)| \le |q_i - f(x_i)| + |f(x_i) - f(x)| < \frac{\varepsilon}{4} + \frac{\varepsilon}{4} = \frac{\varepsilon}{2}.\]
2.9. Therefore \(|p(x) - f(x)| < (1 - \lambda) \frac{\varepsilon}{2} + \lambda \frac{\varepsilon}{2} = \frac{\varepsilon}{2} < \varepsilon\) for all \(x \in [0, 1]\).
2.10. Taking the supremum shows \(d_\infty(p, f) \le \frac{\varepsilon}{2} < \varepsilon\).
3. Second countability:Proof:
3.1. The collection of open balls \(\mathcal{B} = \{B_d(p, r) \mid p \in \mathcal{D}, r \in \mathbb{Q}_+\}\) is a countable collection of open sets.
3.2. Since \(\mathcal{D}\) is dense in the metric space \(\mathcal{C}([0, 1], \mathbb{R})\), \(\mathcal{B}\) forms a topological basis.
3.3. Thus \(\mathcal{C}([0, 1], \mathbb{R})\) is second-countable. Q.E.D.