okay here's a complicated proof: Let $G$ be the group of orientation-preserving symmetries of a regular tetrahedron. The action of $G$ on the set of vertices gives an isomorphism $G \cong A_4$.
The vertices of the regular tetrahedron can be taken to be $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$ this implies that the natural inclusion $G \to SO(3)$ has image contained in $\mathrm{SL}_2(\Bbb Z) \cap SO(3)$. Because they are orthogonal, they preserve the quadratic form $x^2+y^2+z^2$. Reducing mod $3$, we obtain matrices in $\mathrm{SL}_2(\Bbb F_3)$ preserving the quadratic form $x^2+y^2+z^2$, thu…

@TedShifrin Hello. I have been working on Exercise 2.3.1 to 2.3.15 and I am a bit unsure about my solutions to Exercises 2.3.11 and 2.3.13. Should you have the time and inclination to take a look at them, I would appreciate some feedback. My solutions follow.
E.2.3.11. Give an example of a discontinuous function $f:\mathbb{R}\to\mathbb{R}$ having the property that for every $c\in\mathbb{R}$, the level set $f^{-1}(\{c\})$ is closed.
Solution. $f:\mathbb{R}\to\mathbb{R}, f(x)=\mathbbm{1}_{\mathbb{Q}}+x$.

E.2.3.13. Prove that if $\mathbf{f}$ is continuous, then the preimage of every closed set is closed.

Proof. Let $C\subset\mathbb{R}^m$ be a closed set, $f:\mathbb{R}^n\to\mathbb{R}^m$ a continuous function and suppose that $f^{-1}(C)$ is not closed. Then $\mathbb{R}^n\setminus f^{-1}(C)$ is not open so there must be some $\mathbf{x}_0\in\mathbb{R}^n\setminus f^{-1}(C)$ such that $B(\mathbf{x}_0,r)\cap \mathbb{R}^n\setminus (\mathbb{R}^n\setminus f^{-1}(C))=B(\mathbf{x}_0,r)\cap f^{-1}(C)\neq\emptyset$ for every $r>0$.

@TedShifrin Direct proof of Exercise 2.3.13.
Proof. Let $C\subset\mathbb{R}^m$ be a closed set, $f:\mathbb{R}^n\to\mathbb{R}^m$ a continuous function and $\{\mathbf{x}_k\}_{k\in\mathbb{N}}$ a sequence of elements in $\mathbf{f}^{-1}(C)$ which converges to some $\mathbf{a}\in\mathbb{R}^n$.
Then $\{\mathbf{f}(\mathbf{x}_k)\}_{k\in\mathbb{N}}$ is a sequence of elements in $C$ and since $\mathbf{f}$ is a continuous function we have $\mathbf{f}(\mathbf{x}_k)\xrightarrow{k\to\infty}\mathbf{f}(\mathbf{a})$ by Proposition (3.6). Now $\mathbf{f}(\mathbf{a})$ belongs to $C$ since it is …