Equivalence of Definitions of Connected Set (Complex Analysis)

Theorem

The following definitions of the concept of Connected Set in the context of Complex Analysis are equivalent:

Definition 1

$D$ is connected if and only if every pair of points in $D$ can be joined by a staircase contour.

Definition 2

$D$ is connected if and only if every pair of points in $D$ can be joined by a polygonal path all points of which are in $D$.

Proof

$(1)$ implies $(2)$

Let $D$ be a connected set by definition 1.

Then by definition:

Every pair of points in $D$ can be joined by a staircase contour.

But a staircase contour is a polygonal path all of whose points are in $D$.

Thus $D$ is a connected set by definition 2.

$\Box$

$(2)$ implies $(1)$

Let $D$ be a connected set by definition 2.

Then by definition:

Every pair of points in $D$ can be joined by a polygonal path $P$ all points of which are in $D$.

This theorem requires a proof. In particular: It remains to be shown that $P$ can be replaced by a staircase contour. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Thus $D$ is a connected set by definition 1.

$\blacksquare$