On topologies defined by irreducible sets
... is sober) if every closed irreducible set (i.e., a set that is both irreducible and closed) is the closure of a unique singleton. Every sober space is necessarily T0 . The first essential application of sobriety is probably in the characterizing of spaces X which satisfy the following condition: For ...
... is sober) if every closed irreducible set (i.e., a set that is both irreducible and closed) is the closure of a unique singleton. Every sober space is necessarily T0 . The first essential application of sobriety is probably in the characterizing of spaces X which satisfy the following condition: For ...
Introduction to spectral spaces
... Exercise∗ : Suppose C ⊆ Rn is closed and definable. Is C ∈ L ? We approach the question as follows: Let L0 be the lattice of all closed and definable subsets of Rn . So L ⊆ L0 . We want to show ...
... Exercise∗ : Suppose C ⊆ Rn is closed and definable. Is C ∈ L ? We approach the question as follows: Let L0 be the lattice of all closed and definable subsets of Rn . So L ⊆ L0 . We want to show ...
A survey of ultraproduct constructions in general topology
... is an isomorphism. Similarly, if Cop is the opposite of the category C (i.e., same objects, arrows reversed), and if Cop is equipped with ultraproducts, then an object of C is called ultracofinite if it is ultrafinite in Cop . In the setting of concrete categories; i.e., those suitably endowed with ...
... is an isomorphism. Similarly, if Cop is the opposite of the category C (i.e., same objects, arrows reversed), and if Cop is equipped with ultraproducts, then an object of C is called ultracofinite if it is ultrafinite in Cop . In the setting of concrete categories; i.e., those suitably endowed with ...
Compatibility Equations of Nonlinear Elasticity for Non
... compatibility equations will be shown to be equal to N β1 (B), where B is the material manifold, N = dim S (dimension of the ambient space), and β1 (B) is the first Betti number of B , i.e. the dimension of its first homology group with real coefficients H1 (B ; R) or equivalently the rank of its fi ...
... compatibility equations will be shown to be equal to N β1 (B), where B is the material manifold, N = dim S (dimension of the ambient space), and β1 (B) is the first Betti number of B , i.e. the dimension of its first homology group with real coefficients H1 (B ; R) or equivalently the rank of its fi ...
Some descriptive set theory 1 Polish spaces August 13, 2008
... and let O = σ∈S Nσ . O is open in NN since it is a union of open sets. We claim that φ−1 (B (x)) = O (which suffices to prove that φ is continuous). First we show that O ⊆ φ−1 (B (x)). Consider f ∈ O and fix σ ∈ S such that f ∈ Nσ . Since f ∈ Nσ implies that φ(f ) ∈ Vσ , we have that φ(f ) ∈ B (x ...
... and let O = σ∈S Nσ . O is open in NN since it is a union of open sets. We claim that φ−1 (B (x)) = O (which suffices to prove that φ is continuous). First we show that O ⊆ φ−1 (B (x)). Consider f ∈ O and fix σ ∈ S such that f ∈ Nσ . Since f ∈ Nσ implies that φ(f ) ∈ Vσ , we have that φ(f ) ∈ B (x ...
ON SOMEWHAT β-CONTINUITY, SOMEWHAT β
... Theorem 3.4: A function f : (X, τ) → (Y, σ) is hardly β-open if and only if intβ(f–1(A)) = 0/ for each set A ⊂ Y having the property that intβ(A) = 0/ and A containing a nonempty closed set. Proof: Assume f is hardly β-open. Let A ⊂ Y such that intβ(A) = 0/ and let F be a nonempty closed set contain ...
... Theorem 3.4: A function f : (X, τ) → (Y, σ) is hardly β-open if and only if intβ(f–1(A)) = 0/ for each set A ⊂ Y having the property that intβ(A) = 0/ and A containing a nonempty closed set. Proof: Assume f is hardly β-open. Let A ⊂ Y such that intβ(A) = 0/ and let F be a nonempty closed set contain ...
DISCONTINUOUS GROUPS AND CLIFFORD
... 0.5.2, we mention the Auslander conjecture which asserts that the fundamental group π1 of any compact complete affine manifold is virtually solvable (see [Au64], [Mi77], [Ma83] and references therein). In view of Example 0.5.1, this is equivalent to the conjecture that a discrete group Γ is virtually ...
... 0.5.2, we mention the Auslander conjecture which asserts that the fundamental group π1 of any compact complete affine manifold is virtually solvable (see [Au64], [Mi77], [Ma83] and references therein). In view of Example 0.5.1, this is equivalent to the conjecture that a discrete group Γ is virtually ...
