2.2: Open Disks, Open Deleted Disks, and Open Regions

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Definition: Open Disk

The open disk of radius $r$ around $z_0$ is the set of points $z$ with $|z - z_0| < r$, i.e. all points within distance $r$ of $z_0$.

The open deleted disk of radius $r$ around $z_0$ is the set of points $z$ with $0 < |z - z_0| < r$. That is, we remove the center $z_0$ from the open disk. A deleted disk is also called a punctured disk.

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Left: an open disk around $z_0$; right: a deleted open disk around $z_0$

Definition: Open Region

An open region in the complex plane is a set $A$ with the property that every point in $A$ can be be surrounded by an open disk that lies entirely in $A$. We will often drop the word open and simply call $A$ a region.

In the figure below, the set $A$ on the left is an open region because for every point in $A$ we can draw a little circle around the point that is completely in $A$. (The dashed boundary line indicates that the boundary of $A$ is not part of $A$.) In contrast, the set $B$ is not an open region. Notice the point $z$ shown is on the boundary, so every disk around $z$ contains points outside $B$.

2.2.svg — Figure $\PageIndex{1}$: Left: an open region $A$; right: $B$ is not an open region. (CC BY-NC; Ümit Kaya)