Sheaves of Modules
... In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Zmodules. Basic references are [Ser55], [DG67] and [AGV71]. This is a chapter of the Stacks Project, version d59dda8, compiled on Apr 24, ...
... In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Zmodules. Basic references are [Ser55], [DG67] and [AGV71]. This is a chapter of the Stacks Project, version d59dda8, compiled on Apr 24, ...
Supra b-compact and supra b
... Prof: We will show the case when A is supra b-compact relative to X, the other case is similar. Suppose that Ũ = {Uα : α ∈ ∆} is a cover of A ∩ B by supra b-open sets in X. Then Õ = {Uα : α ∈ ∆} ∪ {X − B} is a cover of A by supra b-open sets in X, but A is supra b-compact relative to X, so there e ...
... Prof: We will show the case when A is supra b-compact relative to X, the other case is similar. Suppose that Ũ = {Uα : α ∈ ∆} is a cover of A ∩ B by supra b-open sets in X. Then Õ = {Uα : α ∈ ∆} ∪ {X − B} is a cover of A by supra b-open sets in X, but A is supra b-compact relative to X, so there e ...
Topological Dynamics: Minimality, Entropy and Chaos.
... Given a dynamical system (X , f ) , a set M ⊆ X is called a minimal set if it is non-empty, closed and invariant and if no proper subset of M has these three properties. So, M ⊆ X is a minimal set if and only if (M, f |M ) is a minimal system. A system (X , f ) is minimal if and only if X is a minim ...
... Given a dynamical system (X , f ) , a set M ⊆ X is called a minimal set if it is non-empty, closed and invariant and if no proper subset of M has these three properties. So, M ⊆ X is a minimal set if and only if (M, f |M ) is a minimal system. A system (X , f ) is minimal if and only if X is a minim ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.