Computability of a topological poset

For subspaces X and Y of Q the notation Xh_?Y means that X is homeomorphic to a subspace of Y and X∼Y means Xh_?Yh_?X. The resulting set P(Q)/∼ of equivalence classes is partially-ordered by the relation if Xh_?Y. In a previous paper by the author it was established that this poset is essentially determined by considering only the scattered X⊆Q of finite Cantor-Bendixson rank. Results from that paper are extended to show that this poset is computable.     

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

Let X be a 1-connected CW-complex of finite type and ε_?(X) be the group of homotopy classes of self-equivalences of X which induce the identity on homotopy groups. In this paper, we prove that every finitely generated 2-solvable rational nilpotent group is realizable as ε_?(X) where X is the rationalization of a 1-connected CW-complex of finite type.     

The embeddability ordering of topological spaces

For K a set of topological spaces and X,Y∈K, the notation Xh_⊆Y means that X embeds homeomorphically into Y; and X∼Y means Xh_⊆Yh_⊆X. With , the equivalence relation ∼ on K induces a partial order h_? well-defined on K/∼ as follows: if Xh_⊆Y.For posets (P,P_?) and (Q,Q_?), the notation (P,P_?)?(Q,Q_?) means: there is an injection such that p₀P_?p₁ in P if and only if h(p₀)Q_?h(p₁) in Q. For κ an infinite cardinal, a poset (Q,Q_?) is a κ-universal poset if every poset (P,P_?) with |P|?κ satisfies (P,P_?)?(Q,Q_?).The authors prove two theorems which improve and extend results from the extensive relevant literature.

Theorem 2.2. There is a zero-dimensional Hausdorff space S with|S|=κsuch that(P(S)/∼,h_?)is a κ-universal poset.     

Priestley duality for order-preserving maps into distributive lattices

The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice L of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-ech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; X ^Y denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of L ^P is determined, where P is a poset and L a bounded distributive lattice.     

On Homotopy Types of Alexandroff Spaces

We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. In particular, we characterize pairs of spaces X,Y such that the compact-open topology on C(X,Y) is Alexandroff, give a homotopy type classification of a class of infinite Alexandroff spaces and prove some results concerning cores of locally finite spaces. We also discuss a mistake found in an article of F.G. Arenas. Since the category of T ₀ Alexandroff spaces is equivalent to the category of posets, our results may lead to a deeper understanding of the notion of a core of an infinite poset.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

Duality results for perturbation classes of semi-Fredholm operators

The perturbation classes problem for semi-Fredholm operators asks when the equalities SS(X,Y)=PF₊(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)} and SC(X,Y)=PF_-(X,Y){\mathcal{SC}(X,Y)=P\Phi_-(X,Y)} are satisfied, where SS{\mathcal{SS}} and SC{\mathcal{SC}} denote the strictly singular and the strictly cosingular operators, and PΦ₊ and PΦ₋ denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, SS(Y^*,X^*)=PF₊(Y^*,X^*){\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)} if and only if SC(X,Y)=PF_-(X,Y),{\mathcal{SC}(X,Y)=P\Phi_-(X,Y),} and SC(Y^*,X^*)=PF_-(Y^*,X^*){\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)} if and only if SS(X,Y)=PF₊(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}. Moreover we give examples showing that both direct implications fail in general.     

On dimensionally exotic maps

We call a value y = f(x) of a map f: X → Y dimensionally regular if dimX ≤ dim(Y × f ^?1(y)). It was shown in [6] that if a map f: X → Y between compact metric spaces does not have dimensionally regular values, then X is a Boltyanskii compactum, i.e., a compactum satisfying the equality dim(X × X) = 2dim X ? 1. In this paper we prove that every Boltyanskii compactum X of dimension dim X ≥ 6 admits a map f: X → Y without dimensionally regular values. We show that the converse does not hold by constructing a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.     

Zero-dimensional proximities and zero-dimensional compactifications

We introduce zero-dimensional proximities and show that the poset 〈Z(X),?〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈Π(X),?〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈Π(X),?〉 is isomorphic to the poset 〈B(X),⊆〉 of Boolean bases of X, and derive Dwinger's theorem that 〈Z(X),?〉 is isomorphic to 〈B(X),⊆〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone-?ech compactification of X is a unique up to equivalence extremally disconnected compactification of X.     

Ideals of Priestley powers of semilattices

Let X be a poset and Y an ordered space; X ^Y denotes the poset of continuous order-preserving maps from Y to X with the discrete (respectively, Scott, Lawson) topology. If S is a -semilattice, its ideal semilattice, and T a bounded distributive lattice with Priestley dual space P(T), it is shown that the following isomorphisms hold: Moreover, and sufficient conditions and necessary conditions for the isomorphism to hold are obtained (both necessary and sufficient if S is a distributive -semilattice). Received September 27, 1995; accepted in final form April 6, 1998.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

Local compactness and Hewitt realcompactifications of products II

We generalize and refine results from the author's paper [18]. For a completely regular Hausdorff space X, υX denotes the Hewitt realcompactification of X. It is proved that if υ(X×Y)=υX×υY for any metacompact subparacompact (or m-paracompact) space Y, then X is locally compact. A P(n)-space is a space in which every intersection of less than n open sets is open. A characterization of those spaces X such that υ (X×Y = υX×υY for any (metacompact) P(n)-space Y is also obtained.     

Models for torsion homotopy types

Given an integern>1 and any setP of positive integers, one can assign to each topological spaceX a homotopy universal mapX ^(P,n) →X whereX ^(P,n) is an (n−1)-connected CW-complex whose homotopy groups areP-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces. The authors acknowledge financial aid given by the DGICYT under projects PB93-0581-C02-01 and PB94-0725.     

Remarks on Linear Homeomorphisms of Function Spaces

It is proved that whenever X and Y are completely regular -spaces of pointwise countable type and the spaces C _p(X) and C _p(Y) of real-valued continuous functions on X and Y, respectively, endowed with the topology of pointwise convergence, are linearly homeomorphic, the X is locally compact iff Y is locally compact. This extends the McCoy and Ntantu result.     

An Arithmetic Analogue of Bezout's Theorem

In this paper, we prove two versions of an arithmetic analogue of Bezout's theorem, subject to some technical restrictions. The basic formula proven is deg(V)h(XY)=h(X)deg(Y)+h(Y)deg(X)+O(1), where X and Y are algebraic cycles varying in properly intersecting families on a regular subvariety V _S P _S ^N . The theorem is inspired by the arithmetic Bezout inequality of Bost, Gillet, and Soulé, but improve upon it in two ways. First, we obtain an equality up to O(1) as the intersecting cycles vary in projective families. Second, we generalise this result to intersections of divisors on any regular projective arithmetic variety.     

Isometries and almost isometries between spaces of continuous functions

We characterize the isometries fromC(X) intoC(Y) whereX andY are compact metric spaces. We give necessary and sufficient conditions on an isometry from a subset ofC(X) intoC(Y) to have an extension to the whole space. It is also shown that an almost isometry from the unit ball ofC(X) into the unit ball ofC(Y) is near to an isometry.     

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim_es(X) of X, as defined by Oleg Izhboldin, is dim_es(X)=dim(X)-i(X) +1, where i(X) is the first Witt index of X (i.e., the Witt index of X over its function field).Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X). Our main theorem states that dim_es(X)dim(Y) and that in the case dim_es(X)=dim(Y) the quadric X is isotropic over F(Y).Applying the main theorem to a projective quadric Y, we get a proof of Izhboldins conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X), then dim_es(X)dim_es(Y), and the equality holds if and only if X is isotropic over F(Y). We also solve Knebuschs problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_es(X). To the memory of Oleg Izhboldin     

A Note on the Comedian for Elliptical Distributions

The comedianCOM(X, Y) of random variablesX,Yis a median based robust alternative to the covariance ofXofY. For the bivariate normal case it is known thatCOM(X, Y), standardized by the median absolute deviations ofXandY, is a symmetric, strictly increasing and continuous function of the correlation coefficientρwith range [−1, 1] and can therefore serve as a robust alternative toρ. We show that this result, which is not true in general, extends to elliptical distributions even in the case where moments ofX,Ydo not exist.