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 $D$.
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.
$\blacksquare$