MATH3401

Topology

$\partial Ω =$ boundary of $Ω =$ ${z \in C : z is a boundary point of Ω}$

Note: Interior points belong to $Ω$ , exterior points are in $Ω^{c}$ . What about boundary points??

Consider $Ω = {z \in C : | z | = 1}$ . $\partial Ω = Ω$

Topology

$z_{1}$ interior point

$z_{2}$ exterior point

$z_{3}$ boundary point, $z_{3} \in Ω$

$z_{4}$ boundary point, $z_{4} \notin Ω$

Topology

$Int Ω =$ interior of $Ω$ $= {z \in C : z is an interior point of Ω} .$

$Ext Ω =$ exterior of $Ω$ $= {z \in C : z is an exterior point of Ω} .$

$Ω$ is open if $Ω = Int Ω$ .

$Ω$ is closed if $\partial Ω \subseteq Ω$ .

Topology

Example 1: $Ω_{1} = B_{1} (0)$ $= B_{1} = B$ $= {z : | z | < 1} .$

$Int Ω_{1} = Ω_{1}$ ;

$Ext Ω_{1} = {z : | z | > 1}$ ;

$\partial Ω_{1} = {z : | z | = 1} = S^{1}$ ;

$Ω_{1}^{c} = {z : | z | \geq 1}$ ;

$Ω_{1}$ is open.

Topology

Example 2: $Ω_{2} = \overset{―}{B_{1}} (0)$ $= \overset{―}{B_{1}}$ $= {z : | z | \leq 1} .$

$Int Ω_{2} = Ω_{1}$ ;

$Ext Ω_{2} = Ext Ω_{1}$ ;

$\partial Ω_{2} = \partial Ω_{1}$ ;

$Ω_{2}^{c} = Ext Ω_{2}$ ;

$Ω_{2}$ is closed.

Topology

Example 3: $Ω_{3} = {z : 0 < | z | \leq 1}$ .

$Int Ω_{3} = {z : 0 < | z | < 1}$ ;

$Ext Ω_{3} = Ext Ω_{1}$ ;

$\partial Ω_{3} = S^{1} \cup {0}$ ;

$Ω_{3}^{c} = Ext Ω_{3} \cup {0}$ ;

$Ω_{3}$ is neither open nor closed.

Topology

Note: $Ω_{1}$ is open, $Ω_{1}^{c}$ is closed; $Ω_{2}$ closed, $Ω_{2}^{c}$ open;
$Ω_{3}$ and $Ω_{3}^{c}$ are neither open nor closed.

Only clopen sets (both open and closed) in $C$ are

$C$ and $\emptyset .$

Topology

$*$ $Ω \subseteq C$ is connected if there do not exist nonempty, open, disjoint sets $Ω^{'}$ and $Ω^{''}$ such that $\begin{aligned} Ω \subseteq Ω^{'} \cup Ω^{''} and \\ Ω^{'} \cap Ω \neq \emptyset and Ω^{''} \cap Ω \neq \emptyset . \end{aligned}$

Credits

Lecture notes
Joseph Grotowski

Design, Images & Applets
Juan Carlos Ponce Campuzano

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Complex Analysis