$\mathbb{R}^n \setminus C_r(0)$, $r>0$ connected for n>1

Question

I want to prove that $\mathbb{R}^n \setminus C_r(0)$, $r>0$ is connected for n>1 where $C_r(0)$ denotes the ball around $0$ with radius $r$.

What's the easiest way to show this? Directly or via path connected?

I thought about working with the following:

For $n>2$ the $n-1$ sphere $S^{n-1}$ is connected and $R^n\setminus C_r(0) \longrightarrow\ S^{n-1}: x \mapsto\ \frac{x}{||x||},$ is continuous and surjective.

Is it correct to say that therefore $\mathbb{R}^n \setminus C_r(0)$ has to be connected? Because usually the lemma goes the other way: a continuous image of a connected set is connected.

Or are there other, simpler ways to prove ths?

Answer 1

It is easier to prove that it is path connected. It is not hard to see that if $p$ and $q$ belong to your set, then you can always find a third point $s$ such that both the line segment going from $p$ to $s$ and the line segment going from $s$ to $q$ are subsets of your set.

Answer 2

Let $n \ge 1$ and define ${E_n}^{*} = \mathbb{R}^n \setminus \{\vec 0\}$. It is not difficult to show that the punctured Euclidean space ${E_n}^{*}$ is connected,

Define the unit sphere as $S_{n-1} = \{v \in \mathbb{R}^n \, | \; \; \lVert v \rVert = 1 \}$. The mapping $v \mapsto\ \frac{v}{\lVert v \rVert}$ is a continuous surjection of ${E_n}^{*}$ onto $S_{n-1}$, so the unit sphere is connected.

Recall that if $D$ is a clopen of a topological space $X$ that contains a point $x$ belonging to a connected subspace $G$ of $X$, then $G \subset D$. This is used implicitly several times in what follows.

Exercise: If $B_r(0)$ is a closed ball about the zero vector $\vec 0$ in $\mathbb{R}^n$, show that

$\tag 1 M = \mathbb{R}^n \setminus B_r(0)$

is connected.

Sketch

The sphere $G$ of radius $r + 1$ is a connected subspace of $M$.

$M$ is the union of lines with each one intersecting $G$ in exactly one point (c.f. the polar representation of vectors). Any clopen of $M$ must contain all lines (and line segments/rays) emanating from any point belonging to it.

Let $D$ be any nonempty clopen of $M$ with $\vec v \in M$. By using a scalar factor on $\vec v$, it must be true that $D$ contains a point belonging to $G$. Then it must be true that $G \subset D$. Then it must be true that $M \subset D$. Then it must be true that there is no nontrivial decomposition of $M$ into two clopens. So $M$ is connected.