Open cover

Walls and floor openings covered with e.g. Fire Blanket... 

The family F is called a finite cover if F contains only a finite number of F, subsets. If every open cover of a subset A of a topological space X contains a finite subcover, then the subset A of the topological space X is compact. The compactness property is a generalization of the elementary properties of closed and bounded intervals. 

We will assume for the following that there exist a finite open cover of... 

For the next four lemmas let q be a prime power satisfying gcd(n,q) = 1 and let either S = A be an abelian surface over Fq or let S—>A be a geometrically ruled surface over an elliptic curve A over Fq. In this case we assume that there exist an open cover ([/ ),- of A and isomorphisms a 1( 7i) = /, x Pj over Fq. In both cases we assume that, for all l < n, all the /-division points of A are defined over Fq. All these conditions can be obtained by extending Fq if necessary. Let F be the geometric Frobenius over Fq. We put... 

Poor housekeeping may also be a symptom of other fundamental problems, such as careless operation, frequent temporary repairs, and generally inadequate maintenance. These conditions can result in process leaks, releases, and spills, missing or open covers on equipment and electrical panels, unpainted rusting metal, and nonfunctional gauges and instruments. 

Compactness is usually defined in terms of open covers, and the char acterization we give... 

An obvious example is obtained considering an open ( resp. closed ) subscheme X, of a scheme X correspondingly we obtain an open ( resp. closed ) subfunctor Hom(-,X,) of Hom(-.X). An open cover X, of X defines a cover of Hom(-,X) by open subfunctors... 

Given a scheme 5 and an open cover Uj) of 5, to give a locally free rank n quotient of Vv 0s is equivalent to give one such quotient over every open set Ut so that they patch together on the intersections Ut n Uj. Therefore 6 is a sheaf. 

The fibered product Sj Hom -,5) Gj is evidently represent by the open subscheme of 5 supported on the points where the map E kOs — sr is surjective this proves that Gj is an open subfunctor of G. Since clearly the Sj s cover 5, we also see that the family of subfunctors 6j is an open covering of 6. 

We will assume for the following that there exist a finite open cover (Ui)f=1 of X and local parameters on each of the 17, defined over k. 

Mi)f], while in the twisted structure the Ru component is (Mj)fl. An isomorphism 6 between these two is thus a family of isomorphisms 9l - over Ru. If we interpret (Spec R as an open covering of... 

A subset A of a topological space X is compact if every open cover of A contains a finite subcover. For example, a sphere is compact but if we remove a point from the sphere, the generated punctured sphere is no longer compact. Missing a single point from a sphere may not appear all that important, but due to the resulting lack of compactness, it has important topological consequences. 

There are different cover arrangements to access the interior, such as hinged or pivoting manway, half or full opening covers. Filter cloth maintenance or component adjustments usually require entering the unit in any case, except for small sizes. See Fig. 18-164. 

G with the following property There is an open cover of S (depending on o o... 

Problem. Prove the following converse let X be a prevariety, Ui an affine open covering of X. Let Ri be the coordinate ring of Ui. Then if Ui fl Uj is an affine subset of X with coordinate ring Ri Rj, X is a variety. 

Call this set Yi. From the second form of the intersection it follows that Fj j is an open covering of Y. From the first form of the intersection, it follows that... 

Since q is surjective, this implies that Yi must be an open covering of Y. But now define ... 

Definition 1. A prescheme is a topological space X, plus a sheaf of rings ox on X, provided that there exists an open covering Ua of X such that each pair... 

Proposition 1. Let X be a prescheme over R. If there exists a finite affine open covering Ui, 1 < u < n, of X such that R = T (Ui,ox) is a finitely generated R-algebra, then X is of finite type over R. 

Hi) there exists an affine open covering Ui of X such that f l (Ui) is affine and the map... 

This result is more readily checked privately than proven formally, but it may be instructive to sketch how a morphism F Mg - Mg will induce a morphism / X -4 X2. One chooses an affine open covering Ui = Spec (A ) of X let... 

Definition 4. Let X be a scheme. An -module T is locally free of rank r if there is an open covering [/ of X such that... 

---