Lecture 11
\(\partial \Omega = \) boundary of \(\Omega=\) \[\{z\in \C : z \text{ is a boundary point of } \Omega\}\]
Note: Interior points belong to \(\Omega\), exterior points are in \(\Omega^c\). What about boundary points??
Consider \(\Omega = \{z\in \C : |z|=1\}\). \(\partial \Omega = \Omega\)
\(z_1\) interior point
\(z_2\) exterior point
\(z_3\) boundary point, \(z_3\in \Omega\)
\(z_4\) boundary point, \(z_4\notin \Omega\)
\(\text{Int }\Omega=\) interior of \(\Omega \) \[=\{z\in \C : z \text{ is an interior point of } \Omega\}.\]
\(\text{Ext }\Omega=\) exterior of \(\Omega \) \[=\{z\in \C : z \text{ is an exterior point of } \Omega\}.\]
\(\Omega\) is open if \(\Omega = \text{Int }\Omega\).
\(\Omega\) is closed if \(\partial \Omega \subseteq \Omega\).
Example 1: \(\Omega_1=B_1(0)\) \(=B_1=B\) \[=\{z : |z| \lt 1\}.\]
\(\text{Int }\Omega_1=\Omega_1\);
\(\text{Ext }\Omega_1=\{z : |z| > 1\}\);
\(\partial \Omega_1 = \{z : |z| = 1\} = S^1\);
\(\Omega_1^c = \{z : |z| \geq 1\} \);
\(\Omega_1 \) is open.
Example 2: \(\Omega_2=\overline{B_1}(0)\) \(=\overline{B_1}\) \[=\{z : |z| \leq 1\}.\]
\(\text{Int }\Omega_2=\Omega_1\);
\(\text{Ext }\Omega_2=\text{Ext }\Omega_1\);
\(\partial \Omega_2 = \partial \Omega_1\);
\(\Omega_2^c =\text{Ext }\Omega_2 \);
\(\Omega_2 \) is closed.
Example 3: \(\;\Omega_3 = \{z : 0 \lt |z| \leq 1\}\).
\(\text{Int }\Omega_3=\{z : 0 \lt |z| \lt 1\}\);
\(\text{Ext }\Omega_3=\text{Ext }\Omega_1\);
\(\partial \Omega_3 =S^1 \cup \{0\}\);
\(\Omega_3^c =\text{Ext }\Omega_3 \cup \{0\} \);
\(\Omega_3 \) is neither open nor closed.
Note: \(\Omega_1\) is open, \(\Omega_1^c\) is closed;
\(\Omega_2\) closed, \(\Omega_2^c\) open;
\(\Omega_3\) and \(\Omega_3^c\) are neither open nor closed.
Only clopen sets (both open and closed) in \(\C\) are
\(\C\;\;\) and \( \;\;\emptyset.\)
\(*\) \(\Omega \subseteq \C\) is connected if there do not exist nonempty, open, disjoint sets \(\Omega^{\prime}\) and \(\Omega^{\prime \prime}\) such that \[ \begin{align*} & \Omega \subseteq \Omega^{\prime} \cup \Omega^{\prime\prime } \text{ and } \\ & \Omega^{\prime} \cap \Omega \neq \emptyset\; \text{ and } \;\Omega^{\prime \prime} \cap \Omega \neq \emptyset . \end{align*} \]