Connected Sets in the Complex Plane

Question

I am trying to solve the following problem in Brown and Churchill's Complex Variables text.

Let $S$ be the open set consisting of all points $z$ such that $|z| < 1$ or $|z - 2| < 1$. State why $S$ is not connected.

I have seen different definitions of a connected set, so here is the one in the text.

An open set $S$ is connected if each pair of points $z_1$ and $z_2$ in it can be joined by a polygonal line, consisting of a finite number of line segments, joined end to end, that lies entirely in $S$.

I am having difficulty applying this definition. I tried plotting this set in the complex plane, which amounts to a disk of radius $1$ centered at $0$ and a disk of radius $1$ centered at $2$. The disks do not seem to intersect, but the boundaries, at $z = 1$, do. We can choose points such that we get arbitrarily close to $z = 1$, but the set does not include it. It seems, then, that the fact that the two sets, $|z| < 1$ and $|z-2| < 1$, are disjoint implies that the set cannot be connected since if I take any two points, one in each set, I cannot possibly connect them, as I would need to draw a polygonal line going through $z = 1$, though such a line would not be in the set as $z = 1$ is not.

Is this correct? I am unsure on whether I am getting the intuition here right. This argument, admittedly, does not sound very rigorous.

Thanks in advance.

Answer 1

Let's go with your definition of connectedness. Suppose it is connected, let there be a polygon line $\gamma$ connected two arbirarily chosen points $a,b$ in $|z|<1$ and $|z-2|<1$ respectively. Re-interpret $\gamma$ as a continuous mapping $$\gamma:[0,1]\to\mathbb{C}$$ starting at $a$ and terminating at $b$ (i.e., $\gamma(0)=a,\ \gamma(1)=b$). Let $$t_1=\inf\{t\in[0,1]:\gamma(t)\text{ is not in }|z|<1\}$$ And I think you can take it from here.

Hint: show that $|\gamma(t_1)|=1$ by the continuity of $\gamma$. Hence there's a point on the polygon line that belongs to neither of the two disks.

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Some remarks: you definition of connectedness is called path connectedness (you can replace the polygon line by any continuous path). Another definition of connectedness is

A set is called connected if it cannot be separated into two nonempty parts respectively contained in two disjoint open sets.

In an open set, the second connectedness is equivalent to path connectedness. But generally, path connectedness is stronger than the second.