How can we check if the set is path-connected?

Question

I want to check if the following sets is connected, and path-connected.

A set can be path-connected only if it is connected, or not?

$$\displaystyle{A=\{x\in \mathbb{R}^2 : 1\leq x_1^2+x_2^2\leq 4\}}$$

Let $f: [1, 2]\times[0, 2\pi] \to A$ with $f(r, \theta) = (r\cos\theta, r\sin\theta)$.

The function $f$ is continuous and $[1, 2]\times[0, 2\pi]$ is connected. So the image $A$ is also connected.

How can we check if this is set is path-connected?

Answer 1

As Mees de Vries hinted above, all you have to do is note that $[1, 2] \times [0, 2\pi] \subseteq \mathbb{R}^2$ (where $\mathbb{R}^2$ has the usual topology on it) is path-connected.

And analogously to the theorem that the continuous image of a connected space is connected, there's a theorem that states the continuous image of a path-connected space is path-connected.

So since $f$ is continuous we have $f\left[[1, 2] \times [0, 2\pi]\right] = A$ to be path-connected by the above theorem.