Regarding "Sterntalers", related to the basis of a topology.

Solution 1:

In your attempt you try to show that the $S_\epsilon(x)$ form a basis for some topology on $\mathbb R^n$. See here, section "Definition and basic properties".

But this is not what you are required to do. You must prove that these sets from a basis for the Euclidean topology. This means that you must prove the following:

1. The $S_\epsilon(x)$ are open in the Euclidean topology.

2. For each (Euclidean) open subset $U \subset \mathbb R^n$ and each $\xi \in U$ there exists some $S_\epsilon(x)$ such that $\xi \in S_\epsilon(x) \subset U$.

For notation, let $B_d(c) = \{ a \in \mathbb R^n \mid \lVert a - c \rVert < d \}$ denote the open Euclidean ball with center $c$ and radius $d$.

Note that $S$ is a single fixed Sterntaler which is required to be open, i.e. open in the Euclidean topology. Since $S$ is bounded, we have $S \subset B_R(0)$ for some $R > 0$.

Proof of 1. The maps $\mu_\epsilon : \mathbb R^n \to \mathbb R^n, \mu_\epsilon(a) = \epsilon a$,and $\tau_x : \mathbb R^n \to \mathbb R^n, \tau_x(b) = x + b$, are homeomorphisms. Hence $\tau_x(\mu_\epsilon(S))$ is open in $\mathbb R^n$. But $S_\epsilon(x) = \tau_x(\mu_\epsilon(S))$.

Proof of 2. Let $U \subset \mathbb R^n$ be open and $\xi \in U$. There exists $r > 0$ such that $B_r(\xi) \subset U$. Let $\epsilon = r/R$. Obviously $\xi \in S_\epsilon(\xi)$ because $0 \in S$. We claim that $S_\epsilon(\xi) \subset B_r(\xi)$ which finishes the proof. Consider any element of $\eta \in S_\epsilon(\xi)$. It has the form $\eta = \xi + \epsilon y$ with $y \in S$. We have $\lVert \eta - \xi \rVert = \lVert \epsilon y \rVert = \epsilon \lVert y \rVert < \epsilon R = r$, thus $\eta \in B_r(\xi)$.