Can this be simplified? I am attempting to define what it means a region that has no holes in it because I never seen a definition of this. I am sure that differencial geometry offers one so if you know it please post. (I worked so hard on this eventhough it might seems basic).
Definition: Dualspace is
Definition: The components of dualspace are the sets,
Definition: Dualspace is bounded when its components have a lower and upper bound.
Definition: The component along is the set, . The component along is the set, .
Definition: The interval of existence along the components are the closed intervals,
Definition: A region is dualspace such that any element among the interval of existence is an element of the components of dualspace.
My basic approach is that any vertical lines and horizontal lines are fully contained in a region. Although, the limititation of this definition is that it does include non-simple closed curves. However, it can be generalized to further dimensions.