# MATH3401

Lecture 11

### Topology

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

### Topology

$$z_1$$ interior point

$$z_2$$ exterior point

$$z_3$$ boundary point, $$z_3\in \Omega$$

$$z_4$$ boundary point, $$z_4\notin \Omega$$

#### Topology

$$\text{Int }\Omega=$$ 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 $$\partial \Omega \subseteq \Omega$$.

#### Topology

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.

#### Topology

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.

#### Topology

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.

### Topology

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 $$\;\;\emptyset.$$

### Topology

$$*$$ $$\Omega \subseteq \C$$ 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*}