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1.
Let 2 X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology τF on 2 X has as a subbase all sets of the form {A ∈ 2 X :AV ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2 X :A ?W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τF in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.  相似文献   

2.
Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim F I < ∞ and J is nilpotent. In the latter case, there exist numbers n 0 ∈ ?, C ∈ ?+, and t ∈ ?+ for which gc n (A) < +∞ if nn 0 and gc n (A) ~ Cn t d n as n → ∞, where d = PIexp(A) ∈ ?+. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.  相似文献   

3.
Let G be a connected simple graph, let X?V (G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d T (x) for all xX, where d T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d T (x) ≥ f(x) for all xX if and only if for any nonempty subset S ? X, |N G (S) ? S| ? f(S) + 2|S| ? ω G (S) ≥, where ω G (S) is the number of components of the subgraph induced by S.  相似文献   

4.
We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] G ]?=?[k[X]] G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant fk[X] such that G?×?X f ?→?X f is visible? What if X is locally factorial? What if G is reductive?  相似文献   

5.
We show that for every Borel-measurable mapping Δ: [ω]ωR there exists A ∈ [ω]ω and there exists a continuous mapping Γ: [A]ω → [A]?ω with Γ(X) ? X such that for all X, Y ∈ [A]ω it follows that Δ(X) = Δ(Y) if Γ(X) = Γ(Y). In a sense, this is generalization of the Erdös-Rado canonization theorem  相似文献   

6.
We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A n A n ?1 ) = |G| for each nω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.  相似文献   

7.
For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A of A whose union ?A is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.  相似文献   

8.
9.
The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2n. Then there is a family F, |F| = m, such that F ∈ F, G ? X, | G | > | F | implies G ∈ F and F minimizes the number of pairs (F1, F2), F1, F2F F1 ∩ F2 = ? over all families consisting of m subsets of X.  相似文献   

10.
A proper edge coloring of a graph G is said to be acyclic if there is no bicolored cycle in G.The acyclic edge chromatic number of G,denoted byχ′a(G),is the smallest number of colors in an acyclic edge coloring of G.Let G be a planar graph with maximum degree.In this paper,we show thatχ′a(G)+2,if G has no adjacent i-and j-cycles for any i,j∈{3,4,5},which implies a result of Hou,Liu and Wu(2012);andχ′a(G)+3,if G has no adjacent i-and j-cycles for any i,j∈{3,4,6}.  相似文献   

11.
An ideal J of subsets of a Polish space X has (LK) property whenever for every sequence (An) of analytic sets in X, if lim supnHAnJ for each infinite H then ?nGJ for some infinite G. In this note we present a new class of σ-ideals with (LK) property.  相似文献   

12.
In a recent paper by Engel and Schneider, it was asked if, for every n ? 1, A ∈ τ<n> implies (A+D) ∈ τ<n> for every D = diag[d1, d2,… dn] with di ? 0, 1 ? i ? n. We answer this question in the negative. More precisely, we show that for, any n ? 3, the set
< n>): = {DCn,n:(A+D)∈τ < n> for all A∈τ<n>} is exactly given by
(Gt<n>) = {γIn:γ ? 0}.  相似文献   

13.
For every metric space (X, d) and origin oX, we show the inequality I o (x, y) ≤ 2d o (x, y), where I o (x, y) = d(x, y)/d(x, o)d(y, o) is the metric space inversion semimetric, d o is a metric subordinate to I o , and x, yX \ {o} The constant 2 is best possible.  相似文献   

14.
LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allAAA. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.  相似文献   

15.
We consider existential monadic second-order sentences ?X φ(X) about undirected graphs, where ?X is a finite sequence of monadic quantifiers and φ(X) ∈ +∞ωω is an infinite first-order formula. We prove that there exists a sentence (in the considered logic) with two monadic variables and two first-order variables such that the probability that it is true on G(n, p) does not converge. Moreover, such an example is also obtained for one monadic variable and three first-order variables.  相似文献   

16.
A space (X, T) is called I-Lindelöf [1] if every cover A of X by regular closed subsets of the space (X, T) contains a countable subfamily A′ such that X = ∪{int (A): AA′}. In this work we introduce the class of I-Lindelöf sets as a proper subclass of rc-Lindelöf sets [3]. We study various properties of I-Lindelöf sets and investigate the relationship between I-Lindelöf sets and I-Lindelöf subspaces. We give a new characterization of I-Lindelöf spaces in terms of this type of sets. Also, we study spaces (X, T) in which every I-Lindelöf set in (X, T) is closed.  相似文献   

17.
A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.  相似文献   

18.
《代数通讯》2013,41(5):2021-2037
Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A Γ = ∪ {A : T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I Γ is an ideal in R, II Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R Γ. Also, A Γ is an R-algebra and AA Γ is a closure operation on any set A = {A; A is an R-algebra, R ? A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I Γ and (IA)ΓA and between the R-algebras R Γ and A Γ.

  相似文献   

19.
Let ? be a trace on the unital C*-algebra A and M ? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,QA are idempotents and P ? QM ? , then ?((P ? Q)2n+1) = ?(P ? Q) ∈ R for all nN. Let the isometries UA and A = A*∈ A be such that I+A is invertible and U-AM ? with ?(U-A) ∈ R. Then I-A, I?UM ? and ?(I?U) ∈ R. Let nN, dimH = 2n + 1, the symmetry operators U, VB(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.  相似文献   

20.
We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Kar?owicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some aXY the order-type of aX is different from that of aY.In addition to the chromatic number x(G) of these graphs we study χ κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.  相似文献   

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