Extremal continua: A class of non-separating subcontinua

The notion of a terminal continuum, as defined by D.E. Bennett and J.B. Fugate, is used to introduce extremal continua, a class of non-separating subcontinua of a continuum. An extremal continuum can be characterized as a proper subcontinuum Y of a metric continuum X with the property that Y contains a point of irreducibility of each irreducible subcontinuum of X that meets Y. If Y is an extremal subcontinuum of X, then Y does not separate any subcontinuum of X containing Y; moreover, if Y is a proper subcontinuum of X and Y does not cut any subcontinuum of X containing Y, then Y is extremal in X.     

A note on abundance of certain semigroups of transformations with restricted range

Given a set X and a nonempty Y?X, we denote by T(X,Y) the subsemigroup of the full transformation semigroup on X consisting of all transformations whose range is contained in Y. We show that the semigroup T(X,Y) is right abundant but not left abundant whenever Y is a proper non-singleton subset of X.     

Generalized perfect spaces

Given two Banach function spaces X and Y related to a measure μ, the Y-dual space X^Y of X is defined as the space of the multipliers from X to Y. The space X^Y is a generalization of the classical Köthe dual space of X, which is obtained by taking Y = L^t(μ). Under minimal conditions, we can consider the Y-bidual space X^YY of X (i.e. the Y-dual of X^Y). As in the classical case, the containment X ⊂ X^YY always holds. We give conditions guaranteeing that X coincides with X^YY, in which case X is said to be Y-perfect. We also study when X is isometrically embedded in X^YY. Properties involving p-convexity, p-concavity and the order of X and Y, will have a special relevance.     

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.     

A characteristic property of the exponential distribution

The following characterization of the exponential distribution is given: Under suitable conditions on the random variables X and Y, X is exponentially distributed if and only if E[min{X, Y}]=E(X)P(X<Y).     

The product of two dependent random variables with regularly varying or rapidly varying tails

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie-Gumbel-Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY. We quantify the impact as the limit, as x→∞, of the quotient of Pr(XY>x) and Pr(X^∗Y^∗>x), where X^∗ and Y^∗ are independent random variables identically distributed as X and Y, respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.     

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.     

On products of k-spaces

As is well known, k-spaces are characterized as quotient spaces of locally compact spaces. For a certain k-space X, we give characterizations on X for X×Y to be a k-space for every space Y in some class of k-spaces. Also, for certation k-spaces X and Y, we give characterizations on X and Y for X×Y to be a k-space.     

Coincidence of the Isbell and fine Isbell topologies

Let C(X,Y) be the set of all continuous functions from a topological space X into a topological space Y. We find conditions on X that make the Isbell and fine Isbell topologies on C(X,Y) equal for all Y. For zero-dimensional spaces X, we show there is a space Z such that the coincidence of the Isbell and fine Isbell topologies on C(X,Z) implies the coincidence on C(X,Y) for all Y. We then consider the question of when the Isbell and fine Isbell topologies coincide on the set of continuous real-valued functions. Our results are similar to results established for consonant spaces.     

A second-dual method for C(X) isomorphisms

Let X and Y denote compact Hausdorff spaces, C(X) and C(Y) the sup-norm Banach spaces of continuous real-valued functions on X and Y respectively, and φ: C(X) → C(Y) an onto isomorphism whose bound is less than three. With the aid of second duals, we construct a natural function from X into the set of finite subsets of Y. This provides a new approach to results of Amir and Cambern. We then obtain necessary conditions on X and Y for the existence of a bound-two isomorphism; for instance, the existence of open sets U and V of X and Y respectively such that U and V are homeomorphic and XßU and YßV are homeomorphic.     

Approximately spectrum-preserving maps

Let X and Y be superreflexive complex Banach spaces and let B(X) and B(Y) be the Banach algebras of all bounded linear operators on X and Y, respectively. If a bijective linear map Φ:B(X)→B(Y) almost preserves the spectra, then it is almost multiplicative or anti-multiplicative. Furthermore, in the case where X=Y is a separable complex Hilbert space, such a map is a small perturbation of an automorphism or an anti-automorphism.     

Weak similarities of metric and semimetric spaces

Let (X,d _X ) and (Y,d _Y ) be semimetric spaces with distance sets D(X) and D(Y), respectively. A mapping F:?X→Y is a weak similarity if it is surjective and there exists a strictly increasing f:?D(Y)→D(X) such that d _X =f°d _Y °(F?F). It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F:?X→Y is an isometry if X and Y are ultrametric and compact with D(X)=D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.     

Applications of balanced pairs

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.     

A universal strong law of large numbers for conditional expectations via nearest neighbors

For k_n-nearest neighbor estimates of a regression Y on X (d-dimensional random vector X, integrable real random variable Y) based on observed independent copies of (X,Y), strong universal pointwise consistency is shown, i.e., strong consistency P_X-almost everywhere for general distribution of (X,Y). With tie-breaking by indices, this means validity of a universal strong law of large numbers for conditional expectations E(Y|X=x).     

The regular part of a semigroup of transformations with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$ is the largest regular subsemigroup of T(X,Y) and determines Green??s relations on T(X,Y). In this paper, we show that F(X,Y)?T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S ¹,S). Then we describe Green??s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.     

Disk-like products of continua

In 1975 Hagopian proved that continua X and Y are atriodic and hereditarily unicoherent when the product X×Y is disk-like. In this paper, under the same condition, we prove that X and Y are contractible with respect to every ANR and X and Y are tree-like continua in ClassHW.     

Some upper bounds for density of function spaces

Let Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space Cα(X,Y) is at most iw(X)⋅d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space Ck([0,1],Y) is κ².     

A note on a set-valued extension property

Principal result: Suppose Y is metrizable. Then: (a) if X is metrizable and A⊂X is closed, then every continuous g:A→Y extends to an l.s.c. ψ:X→K(Y); (b) Y satisfies (a) for all paracompact X if and only if Y is completely metrizable.     

Products of symmetric spaces, and k-networks

As is well known, every product of symmetric spaces need not be symmetric. For symmetric spaces X and Y, in terms of their balls, we give characterizations for the product X×Y to be symmetric under X and Y having certain k-networks, or Y being semi-metric.     

Algebras of real-valued continuous functions in product spaces

Let {X_i:iϵI} be an arbitrary family of spaces, we say that the cartesian product X has the approximation property when C(X) coincides with the Algebra on X generated by the functions which depend on one variable. In this paper we study the problem of characterizing topologically when an arbitrary product space has the approximation property. We prove that if X is an uncountable pseudo-ℵ₁-compact P-space, then X×Y has the approximation property if, and only if, X×Y is pseudo-ℵ₁-compact. As a corollary we obtain the following characterization for P-spaces: Let X and Y be P-spaces, then X×Y has the approximation property if, and only if, X or Y is countable or X×Y is pseudo-ℵ₁-compact.