Denote $S = \left \{ (x,y) \in \mathbb{R} ^2 : 1 < x^2 + y^2 < 4 \right \}$. Is $S$ open in $\mathbb{R} ^2$ with the Euclidean topology?
I have tried searching this problem online but find arguments using continuous functions, which we have not discussed in class yet (it is an introduction to Topology class. We are using Munkres and are currently in Chapter 2). We have to use the following definition.
Definition: A subset $U \subset \mathbb{R} ^n$ is open if for any $x \in U$ there exists $\epsilon >0$ such that $B(x,\epsilon ) \subset U$.
I think that $S$ is open, but am not sure on how to rigorously prove so. Perhaps I am overthinking this. My attempt is:
Let $x \in S$ with $0 < \epsilon _i <2$ for some $i$. Then $x \in B(x, \epsilon _i) \subset S$.
No, that attempt does not work. For starters you haven't specified what $\epsilon_i$ is. But for any number you pick, I can find points around which the $\epsilon$-neighborhood is not contained within $S$. For example the point $(1.5,0)$ is in the annulus $S$, but the $\epsilon$-ball around this point of radius $0.6$ is not contained, in case that's what you chose.
For a better approach, you'll need to be more variable. And think radially. For example, in polar coordinates, if a point is at radius $R$, with $1<R<2$. A small disk, centered at point $(r,\theta)$, how close to the origin can it get? How far?