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:Dualspaceis

Definition: Thecomponentsof dualspace are the sets,

Definition: Dualspace isboundedwhen its components have a lower and upper bound.

Definition: Thecomponent alongis the set, . Thecomponent alongis the set, .

Definition: Theinterval of existencealong the components are the closed intervals,

respectively.

Definition: Aregionis dualspace such that any element among the interval of existence is an element of the components of dualspace.

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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.