A sub-vertex is any n or fewer boundaries, where n is the dimension of the space. The boundaries are said to contain the sub-vertex.
A vertex is a sub-vertex, where n boundaries contain the vertex.
A nonempy region and sub-vertex are attached iff there is a nonempty region with sidedness identical to the attached region, except wrt every boundary containing the sub-vertex.
A vertex is on the same side as each of its regions wrt a boundary if the boundary does not contain the vertex.
A vertex is accessible to a boundary dividing a super region iff the vertex can be on either side of the boundary without changing which side of the boundary the super region’s vertices are on.
Theorem:
A vertex is inaccessible iff there is a simplex-super-region by which the vertex is inaccessible to the dividing boundary.
Proof:
If the vertex is inaccessible, adding boundaries will not make it accessible, so on to the only if part.
The entire outside of the given super-region is covered by base and vertex regions of simplex-super-regions, so complete the proof by choosing one the given vertex is in.
A vertex is inaccessible iff there is a simplex-super-region by which the vertex is inaccessible to the dividing boundary.
Proof:
If the vertex is inaccessible, adding boundaries will not make it accessible, so on to the only if part.
The entire outside of the given super-region is covered by base and vertex regions of simplex-super-regions, so complete the proof by choosing one the given vertex is in.
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