Here's a breakdown:
* Region: A specific area or portion of space.
* Unbounded: Not limited by any boundary or constraint.
* Extends infinitely: Continues indefinitely in one or more dimensions.
Examples:
* The entire x-y plane: Extends infinitely in both the x and y directions.
* The region above the curve y = x^2: Extends infinitely upward, but is bounded by the curve below.
* The space outside a sphere: Extends infinitely in all directions away from the sphere.
Contrast with Bounded Regions:
A bounded region is a region that is completely enclosed by a boundary. This boundary can be a curve, surface, or combination of both. For example, a circle, a square, or a sphere are all bounded regions.
Key takeaway: Unbounded regions are characterized by their lack of a finite boundary, allowing them to stretch infinitely in one or more dimensions.