Roughly speaking, a local pospace is a topological space together with an open covering whose elements are endowed with a partial order.
2.
In this way a complex of groups can be canonically associated to the nerve of an open covering by orbifold ( or orbispace ) charts.
3.
To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use �tale coverings instead.
4.
This can be proven in a classical setting by considering the bar as an open covering of the compact topological space \ { 0, 1 \ } ^ \ omega.
5.
Indeed, assume we have two disjoint opens covering A union B . Any neighborhood of 1 includes C, and the same is true of 3, so 1 and 3 must be in the same open set.
6.
More precisely one considers preorders on open subsets and one requires that given any open subset U and any open covering ? of U, the preorder associated with U is'generated'by the preorders associated with each member of ?.
7.
By compactness, there are finitely many opens covering } } such that the closure of is disjoint from and each is an open disc about a boundary point such that in small translations in the direction of the normal vector carry } } into.
8.
That is, for some open covering, there are local representations of this differential form as a logarithmic derivative ( modified slightly with the exterior derivative " d " in place of the usual differential operator " d / dz " ).
9.
The notion of a closed immersion is local in the sense that " f " is a closed immersion if and only if for some ( equivalently every ) open covering X = \ bigcup U _ j the induced map f : f ^ {-1 } ( U _ j ) \ rightarrow U _ j is a closed immersion.
10.
Under the assumption that \ textstyle \ Omega satisfies the strong cone property; that is, there exists a finite open covering \ textstyle \ left \ { O _ { i } \ right \ } of \ textstyle \ partial \ Omega and corresponding cones \ textstyle \ { C _ { i } \ } with vertices at the origin such that \ textstyle x + C _ { i } is contained in \ textstyle \ Omega for any \ textstyle x \ textstyle \ in \ Omega \ cap O _ { i }.
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