Definition:Neighborhood (Complex Analysis)
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It has been suggested that this page or section be merged into Definition:Complex Disk/Open. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code. |
This page is about Neighborhood in the context of Complex Analysis. For other uses, see Neighborhood.
It has been suggested that this page be renamed. In particular: Epsilon Neighborhood (Complex Analysis) To discuss this page in more detail, feel free to use the talk page. |
Definition
Let $z_0 \in \C$ be a complex number.
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
The $\epsilon$-neighborhood of $z_0$ is defined as:
- $\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$
Also known as
A neighborhood in this context is often referred to as an open disk (UK spelling: open disc).
Some sources introduce this concept as $\delta$-neighborhood (that is: delta), but it is the same thing.
Also see
- Complex Plane is Metric Space: this definition is compatible with that of an open $\epsilon$-ball neighborhood in a metric space.
Linguistic Note
The UK English spelling of neighborhood is neighbourhood.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Point Sets: Complex Numbers: $1.$