# 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 $D$.

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

$\blacksquare$