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 $\displaystyle \mathbb{D}\subseteq \mathbb{R}^2$

Definition: Thecomponentsof dualspace are the sets,

$\displaystyle S_x=\{x|(x,y)\in\mathbb{D}\} \mbox{ and }S_y=\{y|(x,y)\in \mathbb{D}\}$

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

Definition: Thecomponent along$\displaystyle x_0\in S_x$ is the set, $\displaystyle S_x^{x_0}=\{y|(x_0,y)\in \mathbb{D}\}$. Thecomponent along$\displaystyle y_0\in S_y$ is the set, $\displaystyle S_y^{y_0}=\{x|(x,y_0)\in \mathbb{D}\}$.

Definition: Theinterval of existencealong the components $\displaystyle S_x^{x_0} \mbox{ and }S_y^{y_0}$ are the closed intervals,

$\displaystyle [\min\{S_x^{x_0}\},\max\{S_x^{x_0}\}]$

$\displaystyle [\min\{S_y^{y_0}\},\max\{S_y^{y_0}\}]$ 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.