Having difficulty finding the differences between a connected set and a simply connected set and a region.
Would be good if someone could inform me and also give an example.
Thanks
Tom
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$\begingroup$A connected set is a set that cannot be divided into two disjoint nonempty open (or closed) sets. Intuitively, it means a set is 'can be travelled' (not to be confused with path connected, which is a stronger property of a topological space - every two points are connected by a curve).
A simply connected set (let me short it to SC for now) is path-connected (already stronger than just connected) and has one of the following (equivalent) properties:
(Topologicaly SC) Every curve (a path between to points) can be shrunk to a point (or to another curve) continuously - i.e. there is an homotopy between any two curves.
(Analytically SC) Every analytic function has an antiderivative, or equivalently - the integral of any such function on closed curves is zero.
(Homologically SC) For any $z\notin U$ and any curve $\gamma \subseteq U$, $Ind_\gamma (z)=0$.
If $U^C = F \cup K$ (disjoint union) such that $K$ is compact and $F$ is closed, then $K = \emptyset$.
Intuitively, simply connected means that "it has no holes". For complex analysis I think definitions 2 and 3 are the most useful.
As for examples, wolfram has a nice one
The third is not connected and not simply connected, and the fourth is connected but not simply connected.
$\endgroup$ $\begingroup$A connected set is a set that cannot be split up into two disjoint open subsets (this of course depends on the topology the set has; for the case of $\mathbb{C}$, this is the same as the Euclidean topology on $\mathbb{R}^2$). Now, a simply connected set is a path-connected set (any two point can be joined by a continuous curve) where any closed path (a loop) that you draw in the space can be continuously shrunk to a point. An example of disconnected set in $\mathbb{C}$ is the union of two disjoint discs. If we call $B_r(z_0) = \{z \in \mathbb{C} \; : \; |z-z_0| < r\}$ then we can consider the disconnected set $B_1(2i) \cup B_1(-i)$.
A couple examples of connected sets are the unit disc $B_1(0)$, and say that annulus $A = \{z \in \mathbb{C} \; : \; 1 < |z| < 2\}$. Now, the disc is simply connected while the annulus is not. Any loop that you can draw in $B_1(0)$ can be continuously shrunk to a point, while there are loops that you can draw in $A$ (say for instance the curve $\varphi:[0,2\pi] \to A$ given by $\varphi(t) = \frac{3}{2} e^{2\pi i t}$) that can't be shrunk to a point.
$\endgroup$ 2 $\begingroup$A connected set is a set which cannot be written as the union of two non-empty separated sets. This is when the set is made only of one-part, if one wants to think of it intuitively.
However, simple-connectedness is a stronger condition. It requires that every closed path be able to get shrunk into a single point (continuously) and that the set be path-connected. This means that it contains no holes and there is a continuous path between any two points of the set. For an open set in $\mathbb{C}^n$, connectedness is equivalent to path-connectedness.
To see why this is not true if there's a hole, imagine a pole and a rope about it (a closed one). When you try to shrink it continuously (without cutting) into a point, the rope eventually hits the pole.
A region is just an open non-empty connected set.
As for examples, a non-connected set is two unit disks one centered at $1$ and the other at $4$. And for a connected set which is not simply-connected, the annulus forms a sufficient example as said in the comment. If the annulus is to be without its borders, it then becomes a region.
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