how to prove that a subset is Path-connected

Question

For example, if you are given the subset S = {a < arg(z-z_0) < b, with 0 < a < b < π} how do you show that it is path connected?

I know, that I need to show that there exists a path from one arbitrary point to another arbitrary point, say a path from z1 to z2, but I am not sure how to do it. Are there any special ways of how to go about this problem?

Answer 1

There are several „methods“ to show that a subset say $X\subseteq \Bbb R^2$ is path-connected. They do generalize to higher dimensions.

• given any two points $x,y\in X$ one may write down a path manually. For example in the case of $\Bbb R^2\setminus \{0\}$ we can use polar coordinates $x=(r\cos \alpha, r\sin \alpha)$ and $y=(s\cos \beta, s\sin \beta)$ and then check that the function $$\begin{array}{rcl} [0,1] & \rightarrow & \Bbb R^2\setminus \{0\}\\ t &\mapsto & (\operatorname{lerp}(r,s,t)\cos(\operatorname{lerp}(\alpha, \beta, t)), \operatorname{lerp}(r,s,t)\sin(\operatorname{lerp}(\alpha, \beta, t))) \end{array}$$ is a well defined continuous function constituting a path from $x$ to $y$. Here $\operatorname{lerp}$ denotes the usual linear interpolation formula $$\begin{array}{rcl} \Bbb R^2\times[0,1]& \rightarrow & \Bbb R\\ (u,v,t) & \mapsto & (1-t)u + tv \end{array}$$
• one might also try to check, whether $X$ is star-connected, meaning that there is a dedicated point $x\in X$ such that for any $y\in X$ there is a straight $xy$-line in $X$. This would be the way to show that $\Bbb R^2, \Bbb B^2, \Delta^2$ etc. are path-connected.
• we can alter the second idea to use a finite zigzag of paths, useful for example to show the connectivity of the handle-bar (two balls joined by a straight line)
• show that $X$ is in the image of a path-connected space under a continuous map.

These are just some immediate approaches. I am sure way more sophisticated proofs of path-connectivity have been used somewhere. But you will come quite far with those I mentioned.