Equivalence of Definitions of Connected Set (Complex Analysis)

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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.

$\blacksquare$