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Let $D \subseteq \C$ be a subset of the set of complex numbers.
$D$ is a region of $\C$ if and only if:
- $(1): \quad$ $D$ is non-empty
- $(2): \quad$ $D$ is path-connected.
Also defined as
Some sources insist that in order for a subset of $\C$ to be a region it must also be open.
- 1973: John B. Conway: Functions of One Complex Variable $III$: Elementary Properties and Examples of Analytic Functions: $\S2$: Analytic Functions: $2.21$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $11.$