The measure of noncompactness (gamma(A)) of A is defined by gamma(A)=inf{delta>0mid mbox{for }A mbox{ there exists a finite cover by sets whose diameter}leqdelta}.
Then there exists an étale cover ( U i ) i = 1,.., l → X such that: (i) H × X U i is a split reductive U i -group scheme, (ii) U i = Spec ( R i ) with R i a normal noetherian domain.
By Lemma 7.2 in [42], since (|N omega)|) is a locally finite polyhedron (thus an ANR), there exists an open cover α of (|N omega)|) such that any two continuous mappings (f,g Zrightarrow|N omega)|) of a space Z that are α-near are homotopic.
Hence, if there exists a mutually disjoint cover set where, an optimal schedule can be found by activating each cover for cycles.
We can thus cover X by affine Zariski open subsets X 1, ⋯, X l where X i = Spec ( A i ) and such that there exists a finite étale cover V i → X i for i = 1,.., l which splits H X i.
Thus τ S, S ′ is an N G ( S ) -torsor which is locally trivial (i.e. there exists a Zariski open cover X = ∪ X i such that τ S, S ′ ( X i ) ≠ ∅ ).
Then (S_{n}(mathbf {A})) is bounded in X. Write (A_{k}) for the kth projection of A onto X, and notice that (alpha _{X}(A_{k}) leq alpha_{infty}(mathbf {A})), for if (varepsilon>0) is such that there exists a finite ε-cover of A, then the kth projections of this cover provide a finite ε-cover (in X this time) of (A_{k}).
A subset W of a space ((X,tau)) is called α-paracompact [15] if, for every open cover v of W in ((X,tau)), there exists a locally finite open cover ξ of W that refines v. Let (F :(X, tau rightarrow(Y, sigma)) be an upper supra-continuous harmonic multifunction from ((X,tau)) into a Hausdorff space ((Y,sigma)).
A subset W of a space ((X,tau)) is called α-paracompact [12] if for every open cover v of W in ((X,tau)) there exists a locally finite open cover ξ of W which refines v. Let (F :(X, tau rightarrow(Y, sigma)) be an upper superharmonic-continuous superharmonic multifunction from ((X,tau)) into a Hausdorff space ((Y,sigma)).
U k } is an open α-Q-cover of X, there exists an α-Q-cover {F 1,F 2,….
