When is a Pixley-Roy hyperspace CCC?

A space X is said to satisfy condition (C) if for every Y?X with |Y|=ω₁, any family $\text{G}$ of open subsets of Y with |$\text{G}$|=ω₁ has a countable network. It is easy to see that if X satisfies condition (C), then its Pixley-Roy hyperspace $\text{F}$[X] is CCC. We show that under MA_ω1 condition (C) is also necessary for $\text{F}$[X] to be CCC, but under CH it is not.     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.     

Neighbourhoods of transitive graphs and GRR's

Let X be a vertex-transitive graph with complement $\text{X}$. We show that if both N, the neighbourhood of a vertex in X, and $\text{N}$, the neighbourhood of a vertex in $\text{X}$, are disconnected, then either X is isomorphic to K₃ × K₃ or both N and $\text{N}$ contain isolated vertices. We characterize the graphs which satisfy this last condition and show in consequence that they admit automorphisms of the form (12)(34). It follows that if X is a GRR for some graph G then at least one of N and $\text{N}$ is connected. (X is said to be a graphical regular representation, or GRR, for G if its automorphism group is isomorphic to G and acts regularly on its vertices.) Using this result we determine those groups generated by their involutions which do not have a GRR. The largest such group has order 18. As a corollary we conclude that all non-abelian simple groups have GRR's.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

Un probleme de partition de l'ensemble des parties a trois elements d'un ensemble fini

Let X be a set of n elements. Let $\text{T}$₃(X) be the set of all triples of X. We define a clique as a set of triples which intersect pairwise in two elements. In this paper we prove that if n?6, the minimum cardinality of a partition of $\text{T}$₃(X) into cliques is $\text{[}\text{1}\text{4}\text{(n?1)}{}^2\text{]}$.     

Markov inequalities on partially ordered spaces

Let X(ω) be a random element taking values in a linear space $\text{X}$ endowed with the partial order ≤; let $\text{G}$₀ be the class of nonnegative order-preserving functions on $\text{X}$ such that, for each g∈$\text{G}$₀, E[g(X)] is defined; and let $\text{G}$₁?$\text{G}$₀ be the subclass of concave functions. A version of Markov's inequality for such spaces in P(X ≥ x) ≤ inf_$\text{G}$0E[g(X)]/g(x). Moreover, if E(X) = ξ is defined and if Jensen's inequality applies, we have a further inequality P(X≥x) ≤ inf_$\text{G}$1E[g(X)]/g(x) ≤ inf_$\text{G}$1g(ξ)/g(x). Applications are given using a variety or orderings of interest in statistics and applied probability.     

On the law of large numbers for stationary sequences

We show that if X_i is a stationary sequence for which $\text{S}{}_{\mathrm{n}}\text{B}{}_{\mathrm{n}}$ converges to a finite non zero random variable of constant sign, where S_n=X₁+X₂+?+X_n and B_n is a sequence of constants, then B_n is regularly varying with index 1. If in addition $\text{Σ}\text{P}\text{(|X}{}_1\text{|>B}{}_{\mathrm{n}}$ is finite, then $\text{E}\text{|X}{}_1\text{|}$ is finite, and if in addition to this X_i satisfies an asymptotic independence condition, $\text{E}\text{X}{}_1\text{≠ 0}$.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

On nagata's problem for paracompact σ-metric spaces

Let (A) be the characterization of dimension as follows: Ind X?n if and only if X has a σ-closure-preserving base $\text{W}$ such that Ind B(W)?n?1 for every W?$\text{W}$. The validity of (A) is proved for spaces X such that(i) X is a paracompact σ-metric space with a scale {X_i} such that each X_i has a uniformly approaching anti-cover, or(ii) X is a subspace of the product ΠX_i of countably many L-spaces X_i, the notion of which is due to K. Nagami.(i) and (ii) are the partial answers to Nagata's problem wheter (A) holds or not for every M₁-space X.     

The metacompactness of spaces with bases of subinfinite rank

Let X be a set. A collection $\text{U}$of subsets of X has subinfinite rank if whenever $\text{V}$ ? $\text{U}$, ∩$\text{V}$≠ø, and $\text{V}$ is infinite, then there are two distinct elements of $\text{V}$, one of which is a subset of the other. Theorem. AT₁space with a base of subinfinite rank is hereditarily metacompact.     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

Let {X_n} be a ?-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {X_n} the conditions for ergodicity will be met if there is a compact set K and an ? > 0 such that $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 — 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? ??}$ whenever x lies outside K and $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ∣ X}{}_{\mathrm{n}}\text{=x}}$ is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ? 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? 0}$ for all x outside K. An application to queueing theory is given.     

L-spaces and S-spaces in P(ω)

We show that CH implies that $\text{P}\text{(ω)}$, when equipped with the Vietoris topology, has a subspace which is an L-space and a subspace which is an S-space. This is an immediate consequence of the following purely combinatorial result: CH implies the existence of an ω₁-sequence 〈x_α: α < ω₁〉 in $\text{P}\text{(ω)}$ such that (1) if α<β<ω₁, then $\text{X}{}_{\mathrm{\beta }}\text{?}{}^1\text{X}{}_{\mathrm{\alpha }}$; (2) if I ?ω₁ is unaccountable, then there are distinct α, β ∈ I with X_β ?X_α.     

A note on a theorem of heath

We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family $\text{C}\text{=?}{}_{\mathrm{m?N}}\text{C}{}_{\mathrm{m}}$ of closed sets, interior-preserving over each member C of which is a countable family {$\text{D}$_n(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈$\text{C}$ and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: D∈$\text{D}$_v(Γ))?U.     

A note on Banach spaces containing c0 or C∞

It is shown that every separable Banach space X containing a subspace isomorphic to c₀ has a subspace Y with basis such that $\text{X}\text{Y}\text{~}\text{c}{}_0\text{C}{}_{\mathrm{\infty }}$ and the latter space has a shrinking basis and an unconditional FDD. Moreover, it is shown that X ⊕ C_∞ has a basis if X has the bounded approximation property.     

A classification theorem for essentially binormal operators

An essentially binormal operator on Hilbert space is an operator which is unitarily equivalent to a 2 × 2 matrix of essentially commuting, essentially normal operators. A natural invariant of essentially binormal operators up to unitary equivalence in the Calkin Algebra is the reducing essential 2 × 2 matricial spectrum. A nonempty compact subset X of the set of 2 × 2 matrices is called hypoconvex, if it is the reducing essential 2 × 2 matricial spectrum of an operator on Hilbert space. The set EN₂(X) is then defined to be the set of all equivalence classes (up to unitary equivalence in the Calkin algebra) of essentially binormal operators whose reducing essential 2 × 2 matricial spectrum coincides with X. The aim of this paper is to prove a result that enables one to compute EN₂(X) in terms of the topological structure of the space $\text{X}\text{?}$ of unitary orbits of X. Indeed, it is shown that for every hypoconvex subset X of the set of 2 × 2 matrices, there exists a natural homomorphism from $\text{Ext}\text{(}\text{X}\text{?}\text{)}$ onto EN₂(X). Also, a six term cyclic exact sequence is obtained, which produces a characterization of the kernel of the above-mentioned homomorphism.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.