H-spectral spaces

Let X be a T₀-space, we say that X is H-spectral if its T₀-compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T₁-spaces the T₀-compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T₁-space X in order to get its Wallman compactification spectral.     

Remainders in compactifications of topological groups

In this paper, we consider the following question: when does a topological group G have a Hausdorff compactification bG with a remainder belonging to a given class of spaces? We extend the results of A.V. Arhangel'skii by showing that if a remainder of a non-locally compact topological group G has a countable open point-network or a locally Gδ-diagonal, then G and the compactification bG of G are separable and metrizable.     

The structure of Valdivia compact lines

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed ℵ₁, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

A solution to a spectral poset problem of Lewis and Ohm

We show that there is a 1-dimensional (countable) non-spectral poset X such that for all x≠y∈X, ↑x∩↑y and ↓x∩↓y are finite subsets. On the other hand, we obtain some sufficient conditions for posets to be spectral.     

Order representability in groups and vector spaces

In the present paper, first we study in a systematic way the numerical representation problem for total preorders defined either on groups or on real vector spaces. Then, we consider groups and real vector spaces equipped with a topology, and analyze the fulfillment of the so-called continuous representability property; the latter meaning that every continuous total preorder defined on the given topological space admits a continuous real-valued order-preserving function. We also explore the analogous cases as above for total preorders that are compatible with the given algebraic structure, looking for real-valued, continuous or not, order-preserving functions that, in addition, are algebraic homomorphisms.     

Representable topologies and locally connected spaces

The objective of this paper is to study certain order-theoretic properties of locally connected topologies. In the main theorem we prove that the assertion that a locally connected topology which satisfies the countable chain condition has the continuous representability property is undecidable in ZFC set theory. Some related problems and generalizations are also included.     

Preorderable topologies and order-representability of topological spaces

We furnish characterizations of topologies that coincide with the lower topology or with the order topology of some total preorder defined on a set. Leaning on these characterizations we introduce some applications to the study of continuous and semicontinuous order-representability properties of topological spaces.     

Infinite distributive laws versus local connectedness and compactness properties

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F-distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.     

Scattered compactifications and the orderability of scattered spaces

A space is defined to be suborderable if it is embeddable in a (totally) orderable space. The length of a scattered space X is the least ordinal a such that X(_a), the ath derived set of X, is empty. It is shown that a suborderable scattered space of countable length is hereditarily paracompact, orderable, and admits an orderable scattered compactification.     

Real dicompact textures

A notion of real compactness for completely biregular bi-T₂ ditopological texture spaces is defined and studied under the name real dicompactness. In particular it is shown that real dicompact spaces are nearly plain ∗-spaces, and an important characterization is presented. Finally the connection of this work with topological and bitopological real compactness is discussed in a categorical setting.     

Compact spaces generated by retractions

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. This class, denoted by R, has been introduced in [M. Burke, W. Kubi?, S. Todor?evi?, Kadec norms on spaces of continuous functions, http://arxiv.org/abs/math.FA/0312013]. Allowing continuous images in the definition of class R, one obtains a strictly larger class, which we denote by RC. We show that every space in class RC is either Corson compact or else contains a copy of the ordinal segment ω₁+1. This improves a result of Kalenda from [O. Kalenda, Embedding of the ordinal segment [0,ω₁] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (4) (1999) 777-783], where the same was proved for the class of continuous images of Valdivia compacta. We prove that spaces in class R do not contain cutting P-points (see the definition below), which provides a tool for finding spaces in RC?R. Finally, we study linearly ordered spaces in class RC. We prove that scattered linearly ordered compacta belong to RC and we characterize those ones which belong to R. We show that there are only 5 types (up to order isomorphism) of connected linearly ordered spaces in class R and all of them are Valdivia compact. Finally, we find a universal pre-image for the class of all linearly ordered Valdivia compacta.     

Remainders of arcwise connected compactifications of the plane

A. Lelek asked which continua are remainders of locally connected compactifications of the plane. In this paper we study a similar problem with local connectedness replaced by arcwise connectedness. (Each locally connected continuum is arcwise connected.) We give the following characterization: a continuum X is pointed 1-movable if and only if there is an arcwise connected compactification of the plane with X as the remainder.     

On Directed Sets and their Suprema

The aim of this paper is to investigate partially ordered real linear topological spaces in which directed sets admit a supremum in their closure. In particular, we point out that this property is intimately related to the normality of the ordering cone and also to the Scott continuity of functionals belonging to the nonnegative polar of the ordering cone. Research of Mohamed Ait Mansour was supported by LACO (Laboratoire d'Arithmétique, Calcul Formel et Optimisation), UMR-CNRS 6090, University of Limoges and Agence Universitaire de la Francophonie.     

A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras

This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety A, the order-theoretic conditions of density and compactness which characterise the canonical extension of (the lattice reduct of) any A∈A have truly topological interpretations. In addition, a particular realisation is presented of the canonical extension of A; this has the structure of a topological algebra nA(A) whose underlying algebra belongs to A. Furthermore, each of the operations of nA(A) coincides with both the σ-extension and the π-extension of the corresponding operation on A, with which a canonical extension is customarily equipped. Thus, in particular, the variety A is canonical, and all its operations are smooth. The methods employed rely solely on elementary order-theoretic and topological arguments, and by-pass the subtle theory of canonical extensions that has been developed for lattice-based algebras in general.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

Closure operators in topological groups related to von Neumann's kernel

We study a family of idempotent categorical closure operators in the category of topological Abelian groups (and continuous homomorphisms) related to the von Neumann's kernel. The prominent role is played by the idempotent closure operator g also related to questions from Diophantine approximations and ergodic theory.