Compact Hausdorff Space Research Articles

Rota's Fubini lectures: The first problem

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables.In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem.The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S(A)⊆[0,1]A of finitely additive probability measures on A (the states of A) coincides with the set of [0,1]-valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γA of S(A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ(A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω,FΩ,P), with FΩ the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on FΩ, there is an MV-algebra A and a state σ of A such that (Ω,FΩ,P)≅(μ(A),Fμ(A),γA(σ)).

Orthogonality of bilinear forms and application to matrices

We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to present an elementary proof of the well-known Bhatia-Šemrl Theorem in the real case.

On some types of functions and a form of compactness via $ \omega _{s} $-open sets

<abstract><p>In this paper, $ \omega _{s} $-irresoluteness as a strong form of $ \omega _{s} $-continuity is introduced. It is proved that $ \omega _{s} $-irresoluteness is independent of each of continuity and irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.</p></abstract>

Additive local multiplications and zero-preserving maps on C(X)

Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties of $X$, we find topological conditions on $X$ that are equivalent to each of the following: 1. every additive local multiplication on $C\left( X\right) $ is a multiplication, 2. every additive local multiplication on $C_{R}\left( X\right) $ is a multiplication, and 3. every additive map on $C\left( X\right) $ that is zero-preserving (i.e., $f\left( x\right) =0$ implies $\left( Tf\right) \left( x\right) =0$) has the form $T\left( f\right) =T\left( 1\right) \operatorname{Re}f+T\left( i\right) \operatorname{Im}f$.

An Expository on Some Nonstandard Compactifications

Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement *X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set of the above mentioned equivalence classes. A compact topology on is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in .

Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

We show that the proximity inductive dimension defined by Isbell agrees with the Brouwer dimension originally described by Brouwer (for Polish spaces without isolated points) on the class of compact Hausdorff spaces. This shows that Fedorchuk?s example of a compact Hausdorff space whose Brouwer dimension exceeds its Lebesgue covering dimension is an example of a space whose proximity inductive dimension exceeds its proximity dimension as defined by Smirnov. This answers Isbell?s question of whether or not proximity inductive dimension and proximity dimension coincide.

Relationship among various Vietoris-type and microsimplicial homology theories

In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M. C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord's and our homology are isomorphic for all compact uniform spaces and that Korppi's and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi's homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord's homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi's homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.

On spaces of *-measures on ultrametric spaces

The notion of * -measure on a compact Hausdorff space is introduced and investigated in a previous publication of the first named author. In the present note we consider the set of all * -measures of compact support on an ultrametric space. An ultrametrization of this set is provided, which determines a functor in the category of ultrametric spaces and non-expanding maps. We prove that this functor is locally non-expanding and preserves the class of complete ultrametric spaces.

A study on D-spaces and the metrizability of compact spaces with property (σ-A)

In this article, we introduce two notions which are called property (σ-A) and property (σ-B). They are generalizations of property (A) and property (B), respectively. Every space with a point-countable base (or σ-NSR pair-base) satisfies property (σ-A). Every space with the Collins-Roscoe property satisfies property (σ-B). We show that every compact Hausdorff space with property (σ-A) is metrizable. Thus some known conclusions can be generalized. This shows that property (σ-A) plays a key role in the metrizability of compact Hausdorff spaces.We show that the properties of property (σ-A) and property (σ-B) are closed under finite products. Every finite product of T1-spaces which satisfy property (σ-B) (property (σ-A), σ-sheltering (F), σ-well-ordered (F)) is hereditarily a D-space. If (X,T) satisfies ω1-sheltering (F), then (X,Tω) is hereditarily a D-space. We show that if a space X satisfies ω1-sheltering (F) and every countable discrete subspace of X is closed, then X is hereditarily a D-space. This gives a partial answer to a question posed by Z.Q. Feng and J.E. Porter in 2015.We finally give examples to show that there exists a space which has property (σ-A) but it does not have a point-countable base and there exists a space which has property (C) but it does not have property (σ-A).

Asymptotic compactness in topological spaces

The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact sets. The purpose of this paper is to understand the mechanism under which a given net of subsets of topological spaces is compact in the asymptotic sense. For this purpose, we introduce the notion of asymptotic compactness for nets of subsets and study the connection with the compactness of the limit sets. In this paper, for a given net of nonempty subsets, we prove that the asymptotic compactness and the property that the limit set is a nonempty compact set to which the net converges from above are equivalent in uniformizable spaces. We also study the sequential version of the notion of asymptotic compactness by introducing the notion of sequentiality of directed sets.

On a variant of Tingley's problem for some function spaces

Let (Ω,A,μ) and (Γ,B,ν) be two arbitrary measure spaces, and p∈[1,∞]. SetS(Lp(μ))+:={f∈Lp(μ):‖f‖p=1;f≥0μ-a.e.}, that is, the positive part of the unit sphere of Lp(μ). We show that every surjective isometry Φ:S(Lp(μ))+→S(Lp(ν))+ can be extended (necessarily uniquely) to an isometric order isomorphism from Lp(μ) onto Lp(ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ,B,ν) is localizable (in particular, when it is σ-finite). On the other hand, we show that for compact Hausdorff spaces X and Y, if Φ is a surjective isometry from the positive part of the unit sphere of C(X) to that of C(Y), then there is a homeomorphism τ:Y→X satisfying Φ(f)(y)=f(τ(y)) for f∈S(C(X))+ and y∈Y.

The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

Abstract In this paper, by using the ideal theory in residuated lattices, we construct the prime and maximal spectra (Zariski topology), proving that the prime and maximal spectra are compact topological spaces, and in the case of De Morgan residuated lattices they become compact T 0 {T}_{0} topological spaces. At the same time, we define and study the reticulation functor between De Morgan residuated lattices and bounded distributive lattices. Moreover, we study the I-topology (I comes from ideal) and the stable topology and we define the concept of pure ideal. We conclude that the I-topology is in fact the restriction of Zariski topology to the lattice of ideals, but we use it for simplicity. Finally, based on pure ideals, we define the normal De Morgan residuated lattice (L is normal iff every proper ideal of L is a pure ideal) and we offer some characterizations.

Stone–Weierstrass theorems for Riesz ideals of continuous functions

Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindel\"of locally compact Hausdorff space are given, and used to prove Stone-Weierstra{\ss}-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory.

Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.

On 2-local diameter-preserving maps between C(X)-spaces

The 2-locality problem of diameter-preserving maps between C(X)-spaces is addressed in this paper. For any compact Hausdorff space X with at least three points, we give an example of a 2-local diameter-preserving map on C(X) which is not linear. However, we show that for first countable compact Hausdorff spaces X and Y, every 2-local diameter-preserving map from C(X) to C(Y) is linear and surjective up to constants in some sense. This fact yields the 2-algebraic reflexivity of isometries with respect to the diameter norms on the quotient spaces.

Corson reflections

A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to ℵ2. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight ℵ1 are Corson compact. We use the Gelfand–Naimark duality, and our results are stated in terms of unital abelian C⁎-algebras.

Causal variational principles in the σ-locally compact setting: Existence of minimizers

Abstract We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler–Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy.

Atoms in the lattice of covering operators in compact Hausdorff spaces

Let Comp be the category of compact Hausdorff spaces with continuous maps. A cover of a space in Comp is an irreducible preimage; equivalent covers are identified. A covering operator (co) is a function c assigning to each X in Comp a cover X←cXcX which is minimum among covers Y of X with Y=cY. The family of all such c is denoted coComp. This is a complete lattice (albeit a proper class), with bottom the identity operator id and top the Gleason (extremally disconnected, projective) cover operator g. Here, we completely determine the atoms (minimal elements above id) in the lattice coComp, show that any c≠id in coComp is above an atom, and show that coComp is not atomic. At the end, we make some remarks about what the present paper does and does not tell us about several other categories related to Comp.

Orthogonality preserving pairs of operators on Hilbert C 0(Z)-modules

Suppose that Z is a locally compact Hausdorff space and are -module maps between Hilbert -modules such that for every , implies . Then there exists a bounded complex-valued function φ on Z that is continuous on and satisfies .

Constructing a coarse space with a given Higson or binary corona

For any compact Hausdorff space K we construct a canonical finitary coarse structure EX,K on the set X of isolated points of K. This construction has two properties:(1)If a finitary coarse space (X,E) is metrizable, then its coarse structure E coincides with the coarse structure EX,K generated by the Higson compactification K of X.(2)A compact Hausdorff space K coincides with the Higson compactification of the coarse space (X,EX,K) if the set X is dense in K and the space K is Fréchet-Urysohn. This implies that a compact Hausdorff space K is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) K is perfectly normal; (ii) K has weight w(K)≤ω1 and character χ(K)<p. Under CH every (zero-dimensional) compact Hausdorff space of weight ≤ω1 is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.