Commutative Algebra
... The Spectrum of a ring [Reference: EGA I, Chapter 0 §2] -2. Compute Spec Z as a set and describe its topology. Let A be any ring. Let X be any topological space. -1. For f ∈ A we define D(f ) := Spec A \ V (f ). Prove that the open subsets D(f ) form a basis of the topology of Spec A. 0. Prove that ...
... The Spectrum of a ring [Reference: EGA I, Chapter 0 §2] -2. Compute Spec Z as a set and describe its topology. Let A be any ring. Let X be any topological space. -1. For f ∈ A we define D(f ) := Spec A \ V (f ). Prove that the open subsets D(f ) form a basis of the topology of Spec A. 0. Prove that ...
NOTES ON GROTHENDIECK TOPOLOGIES 1
... as follows. Let U ∈ T and let {Ui → U } be any covering of U. Consider a tuple of sections f i ∈ F (Ui ) such that the sections agree on the overlap. For any two such tuples of { f i ∈ F (Ui )} and { g j ∈ F (Vj )} (for {Vj → U } another covering of U), we identify these tuples iff there is a common ...
... as follows. Let U ∈ T and let {Ui → U } be any covering of U. Consider a tuple of sections f i ∈ F (Ui ) such that the sections agree on the overlap. For any two such tuples of { f i ∈ F (Ui )} and { g j ∈ F (Vj )} (for {Vj → U } another covering of U), we identify these tuples iff there is a common ...
FiniteSpaces.pdf
... for X to the basis for Y that preserves inclusions and the number of elements that determine corresponding basic sets, hence X and Y determine the same element of M . Conversely, suppose that X and Y have minimal bases {U1 , · · · , Ur } and {V1 , · · · , Vr } that give rise to the same element of M ...
... for X to the basis for Y that preserves inclusions and the number of elements that determine corresponding basic sets, hence X and Y determine the same element of M . Conversely, suppose that X and Y have minimal bases {U1 , · · · , Ur } and {V1 , · · · , Vr } that give rise to the same element of M ...
Compact Orthoalgebras - Susquehanna University
... projections – equivalently, closed subspaces – of a Hilbert space H. Orthomodular lattices and posets have been studied extensively. The standard reference is [9]; for a more recent survey, see [2]. Let us agree to write a ⊕ b for the join of orthogonal elements a and b of an orthocomplemented poset ...
... projections – equivalently, closed subspaces – of a Hilbert space H. Orthomodular lattices and posets have been studied extensively. The standard reference is [9]; for a more recent survey, see [2]. Let us agree to write a ⊕ b for the join of orthogonal elements a and b of an orthocomplemented poset ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.