Category:Regions

This category contains results about Regions.
Definitions specific to this category can be found in Definitions/Regions.

Metric Space

Let $M = \struct {A, d}$ be a metric space.

A region of $M$ is a subset $U$ of $M$ such that $U$ is:

$(1): \quad$ non-empty

$(2): \quad$ path-connected.

Complex

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.

Region in the Plane

The usual usage of region is in the real number plane or complex plane.

A point set $R$ in the plane is a region if and only if:

$(1): \quad$ Each point of $R$ is the center of a circle all of whose elements consist of points of $R$

$(2): \quad$ Each point of $R$ can be joined by a curve consisting entirely of points of $R$.