Restricted versions of the Tukey-Teichmüller Theorem that are
... The Tukey-Teichmüller Theorem seems tailor-made for certain applications. For example, it can be used to prove that every vector space has a basis by simply noting that the property of being linearly independent has finite character. Another nice application is a proof of the Alexander Subbase Theo ...
... The Tukey-Teichmüller Theorem seems tailor-made for certain applications. For example, it can be used to prove that every vector space has a basis by simply noting that the property of being linearly independent has finite character. Another nice application is a proof of the Alexander Subbase Theo ...
1. Affinoid algebras and Tate`s p-adic analytic spaces : a brief survey
... Definition-Proposition 1.14. Admissible opens and admissible covering form a G-topology on Max(A). U 7→ O(U ), for U ⊂ X an affinoid subdomain, extends uniquely to a sheaf of Qp -algebra on Max(A) for this G-topology above. The obtained locally ringed G-space is the affinoid Sp(A). Note that if (X, ...
... Definition-Proposition 1.14. Admissible opens and admissible covering form a G-topology on Max(A). U 7→ O(U ), for U ⊂ X an affinoid subdomain, extends uniquely to a sheaf of Qp -algebra on Max(A) for this G-topology above. The obtained locally ringed G-space is the affinoid Sp(A). Note that if (X, ...
α-Scattered Spaces II
... Proposition 2.16 If X is α-scattered, then P(X) = LC(X, τ α ), where τ α = τ ∗ (N (τ )). 2 Recall that a family W of subsets of a topological space (X, τ ) is chain-countable [16] if every disjoint subfamily of W is countable. It is proved in [16] that if the family of all open non-meager sets of a ...
... Proposition 2.16 If X is α-scattered, then P(X) = LC(X, τ α ), where τ α = τ ∗ (N (τ )). 2 Recall that a family W of subsets of a topological space (X, τ ) is chain-countable [16] if every disjoint subfamily of W is countable. It is proved in [16] that if the family of all open non-meager sets of a ...
2008 Final Exam Answers
... Read each question carefully. Organize your answers clearly. Write your answers using complete sentences starting on a right hand page of the blue book. Problem 1. (1) Prove that (0, 1] is not a compact subspace of R. Ans: (1/n, 1] is a cover that does not have any finite refinement (or part (2)). ( ...
... Read each question carefully. Organize your answers clearly. Write your answers using complete sentences starting on a right hand page of the blue book. Problem 1. (1) Prove that (0, 1] is not a compact subspace of R. Ans: (1/n, 1] is a cover that does not have any finite refinement (or part (2)). ( ...
Document
... if f(x1) and f(x2) are distinct whenever x1 and x2 are distinct points of Df. Example: The linear map f(x,y)=(x-y, x+y) is one-to-one in the domain Df=ℝ2. Example: The map f(x,y)=(x2-y2, 2xy) is not one-to-one in the domain Df=ℝ2, since f(x,y)=f(-x,-y). Definition: A function f may fail to be one-to ...
... if f(x1) and f(x2) are distinct whenever x1 and x2 are distinct points of Df. Example: The linear map f(x,y)=(x-y, x+y) is one-to-one in the domain Df=ℝ2. Example: The map f(x,y)=(x2-y2, 2xy) is not one-to-one in the domain Df=ℝ2, since f(x,y)=f(-x,-y). Definition: A function f may fail to be one-to ...
LINEAR DEPENDENCE OF POWERS OF LINEAR FORMS Andrzej
... Remark. The statement (c) in Corollary 2.2 was proved by A. BiałynickiBirula and A. Schinzel [1, Theorem 2] for fields of characteristic 0 or greater than d. Moreover, they showed an example that the statement is no longer true when r = d(n − 1) + 2 ≤ #K + 1. The lower bound for the difference of tw ...
... Remark. The statement (c) in Corollary 2.2 was proved by A. BiałynickiBirula and A. Schinzel [1, Theorem 2] for fields of characteristic 0 or greater than d. Moreover, they showed an example that the statement is no longer true when r = d(n − 1) + 2 ≤ #K + 1. The lower bound for the difference of tw ...
Distances between the conjugates of an algebraic number
... Theorem 1.1. Let K be a number field of degree 3, and Σ a subset of {1, 2, 3}. (i) Suppose that either Σ = {1, 2, 3}, or K is totally real and |Σ| = 2, or Σ = {i, j} where ξ 7→ ξ (i) and ξ 7→ ξ (j) are complex conjugate. Then κ(Σ) = 2. (ii) Suppose that Σ = {i, j}, where one of the embeddings ξ 7→ ξ ...
... Theorem 1.1. Let K be a number field of degree 3, and Σ a subset of {1, 2, 3}. (i) Suppose that either Σ = {1, 2, 3}, or K is totally real and |Σ| = 2, or Σ = {i, j} where ξ 7→ ξ (i) and ξ 7→ ξ (j) are complex conjugate. Then κ(Σ) = 2. (ii) Suppose that Σ = {i, j}, where one of the embeddings ξ 7→ ξ ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.