Regarding "Sterntalers", related to the basis of a topology. Exercise. A "Sterntaler" S is a bounded subset of $\Bbb R^n$ which has the following two properties:
$\hspace{2cm} (i)$ $0 \in S$
$\hspace{2cm} (ii)$ $rx \in S, \forall x \in S \text{ and } r \in [0,1]$
For any open Sterntaler $S$, define $S_\epsilon(x) = \{x + \epsilon y: y \in S\}$, $x \in \Bbb R^n$, $\epsilon > 0$. Show that this sets form a basis for the Euclidian topology.
My attempt. Firsly, we prove that $\forall x \in \Bbb R^n, \exists A \in S_\epsilon: x \in A$. To do so, it suffices to take $A = S_\epsilon(x)$ since $x \in S_\epsilon(x)$, for any $\epsilon > 0$ (by definition of a sterntaler, $0 \in S$ and so, all we have to do is take the case where $y=0$). Secondly, we have to prove the following:
\begin{equation*}
\text{ If }x \in A_1 \cap A_2 \Rightarrow x \in A_3 \subseteq A_1 \cap A_2, \text{ where } A_i \in S_\epsilon, \quad \forall i \in [3]
\end{equation*}
I am having some trouble doing this part of the problem, and I feel like it is majorly because I am having trouble identifying the diferent sets $S_\epsilon$ we can have. For example, to prove what's wanted, I would assume there is a $x$ in both $A_1$ and $A_2$, but I am having trouble defining the equatioons for $A_1$ and $A_2$, i.e., we will have different $x's$ (in the original definition), different $y's$ and different $\epsilon's$? Any help would be apreciatted.
 A: 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:

*

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


*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 $\eta \in B_r(\xi)$.
