Appendix H — Homework 7

Exercise H.1 Let \(X \subseteq \mathbb{R}^n\) be a topological space.

  1. Show that if \(S \subseteq X\) is closed and dense in \(X\), then \(S = X\).

  2. If \(f: X \to \mathbb{R}^m\) is continuous and \(S \subseteq X\) is dense in \(X\), show that we can compute the value of \(f(x)\) for any \(x \in X\) just by knowing values of the from \(f(s)\) for \(s \in S\). Conclude that two continuous functions \(f_1, f_2: X \to \mathbb{R}^m\) that agree on a dense subset are equal everywhere.

Exercise H.2  

  1. Show that if \(H \leq \mathbb{R}\) contains a limit point, then \(H\) is dense in \(\mathbb{R}\).

    Hint: Start with \(x \in \mathbb{R}\); we want to find a sequence in \(H\) converging to \(x\). Notice that we can’t directly apply Proposition 10.4 because we don’t know that \(x \in H\); however, we know \(0 \in H\), since \(H \leq \mathbb{R}\). Try using the Archimedean principle!

  2. Show that if \(H \leq \mathbb{T}\) contains a limit point, then \(H\) is dense in \(\mathbb{T}\).

    Note: The setup for \(\mathbb{T}\) is similar to \(\mathbb{R}\)… except elements of \(\mathbb{T}\) all have the same absolute value! Instead, we can use their angle (i.e., their argument). Be careful: arguments do not have a global notion of ordering, but we do get a global notion of distance \(d(z,w) \coloneqq |\arg(zw^{-1})|\) between any two points \(z, w \in \mathbb{T}\) in terms of shortest arclength.

  3. Show that if a non-trivial subgroup \(H\) of \(\mathbb{T}\) has infinitely many elements, it must contain a limit point. Conclude that the only closed infinite subgroup of \(\mathbb{T}\) is \(\mathbb{T}\) itself.

    Hint: Find a limit point of \(H\) (via compactness, geometry and the pigeonhole principle, or some other approach), then use part (c).

Exercise H.3  

  1. Show that the only closed, proper, non-trivial subgroups of \(\mathbb{R}\) are cyclic of the form \(r \mathbb{Z}\) for \(r \in \mathbb{R}^+\).

    Hint: An \(H \leq \mathbb{R}\) satisfying these properties cannot be dense by Exercise H.1(a), or it would not be proper. Use this along with Exercise H.2(b) to show that \(H\) must have a smallest positive element, \(r > 0\), and argue that \(H = r\mathbb{Z}\).

  2. Show that the only closed, proper, non-trivial subgroups of \(\mathbb{T}\) are cyclic of the form \[ \mathcal{C}_{n} = \left\{ z \in \mathbb{C}\mid z^n = 1 \right\} = \langle e^{2\pi i/n} \rangle, \quad \text{ for some } n \in \mathbb{Z}^+. \]

    Hint: Similarly, prove that \(H\) must have an element \(\zeta\) of smallest positive argument. Then show \(H = \langle \zeta \rangle\).

Exercise H.4 (Bonus) Let \(G = \left\{ (\alpha, \beta) \mid \alpha > 0 \right\} \subset \mathbb{R}^2\), where \((\alpha, \beta) \cdot (\gamma, \delta) \coloneqq (\alpha \gamma, \alpha \delta + \beta)\). Consider \[ f \mapsto \int_{-\infty}^\infty \int_{0}^\infty f(\alpha,\beta) \frac{1}{\alpha^2} \mathop{}\!\mathrm{d}{\alpha} \mathop{}\!\mathrm{d}{\beta}, \] the integral of Example 10.18.

  1. Show the above integral is left-invariant.

  2. Show the above integral is not right-invariant.