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
\[
\begin{align*}
& \Omega \subseteq \Omega^{\prime} \cup \Omega^{\prime\prime } \text{ and } \\
& \Omega^{\prime} \cap \Omega \neq \emptyset\; \text{ and } \;\Omega^{\prime \prime} \cap \Omega \neq \emptyset .
\end{align*}
\]