Definition:Region

Definition

Metric Space

Let $M = \left({A, d}\right)$ be a metric space.

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

$(1): \quad$ non-empty

$(2): \quad$ open, and

$(3): \quad$ path-connected.

Region in the Plane

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

A point set in the plane is called a region iff:

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

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

Interior

The boundary of a region separates its interior from the exterior.

The interior consists of the points of the plane which are the elements of the region.

Such points are called interior points of the region.

It is "usual" that the interior is the "smaller bit" which is visually apparently on the inside as it appears on the page or screen, but this is of course not necessarily the case.

Also see the definition of interior and boundary from a topological perspective.

Bounded

A region in the the plane is bounded if there is a circle in the plane which encloses it.

Also see the definition of bounded in the context of a metric space.

Open Region

An open region is a region without its boundary, i.e. the interior of such a region.