On large cohomological dimension and tautness

Abstract

AbstractWe prove that for a non-discrete spaceX, the inequality DimL(X) ≥ dimL(X) + 1 always holds if (i)Xis paracompact and each point isGδ, or (ii)Xis a completely paracompact Moritak-space. Consequently, ifXis a non-discrete completely paracompact space in which each point is aGδ-set or it is also a Moritak-space then, the equality DimL(X) = dimL(X) + 1 always holds. We apply this equality to show that for such a spaceXthere exists a pointx ∈ Xand a family ϕ of supports onXsuch that {x} is not ϕ-taut with respect to sheaf cohomology. This generalizes a corresponding known result forRn. We also discuss the usual sum theorems for this large cohomological dimension; the finite sum theorem for closed sets is proved, and for all others, counter examples are given. Subject to a small modification, however, all of the sum theorems hold for a large class of spaces.