Stone-Cech Compactification Research Articles

On extension of continuous mappings, topological products and retracts

One of the fundamental concepts of the extension theory developed in the works of A. N. Dranishnikov and E. V. Shchepin is the concept of relation of extensiveness, which was introduced by K. Kuratowski. We describe the behavior of the relation of extensiveness with respect to certain types of subspaces of topological products and spaces, which are the Stone–Cech compactifications of the discrete sums of the Hilbert cube.

Multiplication of convex sets in $C(K)$ spaces

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. We study this for a special class of closed convex sets, namely closed intervals. Our main interest is in correlations between properties of the product of order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space K is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on K is needed, can be seen from the fact that, if K is the Stone–Cech compactification of N, then the product of two intervals in C(K) with continuous boundary functions is always an interval. For any K, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type. Mathematics Subject Classification: primary 52A05; secondary 46J10, 46E15.

Finite Embeddability of Sets and Ultrafilters

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Cech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.

On maps between Stone-Cech compactifications induced by lattice homomorphisms

Broverman has shown that if X and Y are Tychonoff spaces and t:Z(Y)?Z(X) is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map ?: ?X ? ?Y induced by t. In this note we expound this idea and supplement Broverman?s results by first showing that this phenomenon holds in the category of completely regular frames. Among results we obtain, which were not considered by Broverman, are necessary and sufficient conditions (in terms of properties of the map t) for the induced map ? to be (i) the inclusion of a subspace, (ii) surjective, and (iii) irreducible. We show that if X and Y are pseudocompact then t pulls back z-ultrafilters to z-ultrafilters if and only if cl?X t(Z) = ?? [cl?YZ] for every Z ? Z(Y) if and only if t is ?-homomorphism.

The Stone–Čech Compactification of Groupoids

Let G be a discrete groupoid. Consider the Stone–Cech compactification βG of G. We extend the operation on the set of composable elements G (2) of G to the operation * on a subset (βG)(2) of βG×βG such that the triple (βG, (βG)(2), *) is a compact right topological semigroupoid.

Filters on topological groups

We use filter representations of the \({\mathcal {LUC}}\)-compactification of a topological group \(G\) and the Stone–Cech compactification of \(G_d\) to describe the continuous homomorphism \(\pi :\beta G_d \rightarrow G^{{\mathcal {LUC}}}\). We apply this method to study algebraic properties of \(G^{{\mathcal {LUC}}}\).

Left large subsets of free semigroups and groups that are not right large

There are several notions of size for subsets of a semigroup $$S$$ that originated in topological dynamics and are of interest because of their combinatorial applications as well as their relationship to the algebraic structure of the Stone–Cech compactification $$\beta S$$ of $$S$$ . Among these notions are thick sets, central sets, piecewise syndetic sets, IP sets, and $$\Delta $$ sets. Two related notions, namely $$C$$ sets and $$J$$ sets, arose in the study of combinatorial applications of the algebra of $$\beta S$$ . If the semigroup is noncommutative, then all of these notions have both left and right versions. In any semigroup, a left thick set must be a right $$J$$ set (and of course a right thick set must be a left $$J$$ set). We show here that for any free semigroup or free group on more than one generator, there is a set which satisfies all of the left versions of these notions and none of the right versions except $$J$$ . We also show that for the free semigroup on countably many generators, there is a left $$J$$ set which is not a right $$J$$ set.

New types of completeness in metric spaces

This paper is devoted to introduce and study two new properties of completeness in the setting of metric spaces. We will call them Bourbaki-completeness and cofinal Bourbaki-completeness. These notions came from new classes of generalized Cauchy sequences appearing when we try to characterize the so-called Bourbaki-boundedness in a similar way that Cauchy sequences characterize the totally boundedness. We also study the topological problem of metrizability by means of a Bourbaki-complete or a cofinally Bourbaki-complete metric. At this respect, we obtain results in the line to the classical Cech theorem about the complete metrizability of a metric space X in terms of its Stone-Cech compactification beta X. Finally we give some relationships between these kinds of completeness and some properties related to paracompactness and uniform paracompactness in the framework of metrizable spaces.

Measures and Dirichlet Forms Under the Gelfand Transform

Using the standard tools of Daniell–Stone integrals, Stone–Cech compactification, and Gelfand transform, we show explicitly that any closed Dirichlet form defined on a measurable space can be transformed into a regular Dirichlet form on a locally compact space. This implies existence, on the Stone–Cech compactification, of the associated Hunt process. As an application, we show that for any separable resistance form in the sense of Kigami there exists an associated Markov process. Bibliography: 29 titles.

On the question of representation of ultrafilters in a product of measurable spaces

We study representations of ultrafilters of broadly understood measurable spaces that are realized by means of generalized Cartesian products. The structure of the arising ultrafilter space is established; in more traditional measurable spaces, this structure reduces to the realization of a Stone compact space in the form of a Tychonoff product. The developed methods can be applied to the construction of extensions of abstract reachability problems with constraints of asymptotic nature; in such extensions, ultrafilters can be used as generalized elements, which admits a conceptual analogy with the Stone-Cech compactification. The proposed implementation includes the possibility of using measurable spaces, for which the set of free ultrafilters can be described completely. This yields an exhaustive representation of the corresponding Stone compact space for measurable spaces with algebras of sets. The present issue is devoted to I.I. Eremin’s jubilee; the author had many discussion with him on very different topics related to mathematical investigations, and the discussions inevitably led to a deeper understanding of their essence. The author appreciates the possibility of such communication and is grateful to Eremin, who contributed significantly to the development of the mathematical science and education in the Urals.

A homological characterization of topological amenability

Generalizing Block and Weinberger’s characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of G on its Stone-Cech compactification we obtain a homological characterisation of exactness of the group.

Ultracompleteness of hyperspaces of compact sets

A space X is called ultracomplete if it has countable character in its Stone–Cech compactification βX. A space X is called almost locally compact if the set of all points at which X is not locally compact is contained in a compact set of countable outer character. For a given Tychonoff space X let 2X be the hyperspace of all nonempty compact subsets of X endowed with the Vietoris topology. We prove that 2X is almost locally compact if and only if X is locally compact. We also prove that for a countably compact ultracomplete space X the hyperspace Fn(X)={K∈2X∣K has at most n points} is also countably compact ultracomplete for every natural number n. We also analyse ultracompleteness of Fn(X) and 2X.

On totally bounded universal algebras

In the class of topological algebras of a given signature the notions of totally boundedness and a-pseudocompactness are introduced. A topological algebra is totally bounded if it is a subalgebra of a compact algebra. The general properties of totally bounded algebras are studied. The compactifications of topological algebras are investigated too. In particular, the problem of the continuous extension of the operation on the Stone-Cech compactification is studied.

Syndetic sets and amenability

We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak ∗ * -closed convex hull of the two-sided translates of every bounded function on an infinite discrete semigroup contains a constant function. Our proofs use the algebraic properties of the Stone-Cech compactification.

A CHARACTERIZATION OF COMPACTIFICATIONS OF LOCAL COMPACT HAUSDORFF SPACE

It is known that different compactification methods are applied to different spaces. In this paper, it is examined some relations among spaces obtained by being applied different compactification methods (One-point compactification, Stone–Cech compactification, Wallman compactification, Fan–Gottesman compactification) to local compact Hausdorff space and these compactifications are characterized with the inequality αX ≅ wX ≅ X* ≤ ⋯ ≤ βX by the addition of lindelöf conditions to studied space. Moreover, it is found maximum and minimum element of the class of all compactifications of the space. We also give an example to illustrate how to apply mathematical result to an application of compactification in physics.

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Cech compactification βS is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient \(\ensuremath {\mathcal {C}_{b}}(S)/\ensuremath {\mathcal {LMC}}(S)\) contains a linear isometric copy of l∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.

The closure of the smallest ideal of an ultrafilter semigroup

Let S be a discrete semigroup, let β S be the Stone-Cech compactification of S, and let T be a closed subsemigroup of β S. We characterize ultrafilters from the smallest ideal K(T) of T and from its closure c l K(T). We show that, for a large class of closed subsemigroups of β S, c l K(T) is not an ideal of T. This class includes the subsemigroups 0+⊂βℝ d and ℍ κ ⊂β(⊕ κ ℤ2).

Hausdorff Nonstandard Extensions

We introduce the notion of Hausdorff extension of an arbitrary set X and we study the connections with the Stone-Cech compactification \beta X of the discrete space X. We characterize those Hausdorff extensions that satisfy the “transfer principle” of nonstandard analysis, and we investigate the consistency strength of their existence.

Intergenerational Choice: A Paradox and a Solution

In the quest for balanced criterion for the problem of intergenerational choice we explore the role played by different groups of generations, which are represented by the Cech-Stone compactification of the naturals.

The modal logic of $${\beta(\mathbb{N})}$$

Let \({\beta(\mathbb{N})}\) denote the Stone–Cech compactification of the set \({\mathbb{N}}\) of natural numbers (with the discrete topology), and let \({\mathbb{N}^\ast}\) denote the remainder \({\beta(\mathbb{N})-\mathbb{N}}\). We show that, interpreting modal diamond as the closure in a topological space, the modal logic of \({\mathbb{N}^\ast}\) is S4 and that the modal logic of \({\beta(\mathbb{N})}\) is S4.1.2.