WALLMAN REMAINDERS OF REGULAR SPACES

ABSTRACT

This paper shows that the only Hausdorff spaces which can occur as Wallman remainders of Regular spaces are themselves completely regular. This is in contrast to the previously known result that any T1 space can occur as a Wallman remainder.     

SPACES WITH WALLMAN EQUIVALENT INTERMEDIATE SPACES

Abstract

If every infinite closed subset of the Wallman compactification, WX, of a space X must contain at least one element of X, then for any space Y intermediate between X and WX the Wallman compactification WY is homeomorphic to WX. This extends a property which characterizes normality inducing spaces. In the case where X is not normal, however, this is not a characterization, since there are nonnormal spaces for which all intermediate spaces are Wallman equivalent, but have infinite closed subsets contained in WX/X.     

REAL COMPACTIFICATIONS THROUGH ZERO-SET SPACES

Abstract

The Alexandroff (= zero-set) spaces were introduced in [l] as the “completely normal spaces”, and have been studied in a number of more recent papers. In this paper we unify the theory of Wallman realcompactifications via the Alexandroff bases and introduce the realcompactfine Alexandroff spaces as particularly relevant to their investigation. These latter spaces are defined analogously to the A-c uniform spaces which are based on a construction of A.W. Hager [25].     

PROXIMAL LIMIT SPACES

Abstract

The proximal limit spaces are introduced which fill the gap arising from the existence of proximity spaces, uniform spaces, and uniform limit spaces. It is shown that the proximal limit spaces can be considered as a bireflective subcategory of the topological category of uniform limit spaces. A limit space is induced by a proximal limit space if and only if it is a S₁-limit space.     

SOME RESULTS CONCERNING THE WALLMAN COMPACTIFICATION

Abstract

The first part of this paper surveys results and open questions about categories of T₁-spaces on which the wallman compactification induces an epireflection. The second part proves results on spaces whose Wallman remainder is Hausdorff.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

THE WALLMAN COMPACTIFICATION OF FRAMES

Abstract

We show that a B-conjunctive frame L, where B is a normal base for L gives rise to a strong inclusion on L and therefore a compactification of L. The resulting compact regular frame corresponds to the quotient frame obtained by Johnstone in his construction of the Wallman compactification for frames. It is also shown that, in the presence of pseudocompactness the Wallman compactification and the Wallman realcompactification coincide.     

On the role of finite,hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces

We consider how properties of the bonding maps of the inverse spectrum determine properties of the inverse limit. Specifically, we study the limits of inverse spectra of finite T₀-spaces with bonding maps which are either chaining or normalizing. We will show that if the bonding maps are normalizing, then the inverse limit is a normal T₀-space, and therefore, its Hausdorff reflection is its subset of specialization minimal elements. If the maps are chaining, then the inverse limit is a completely normal spectral space; such spaces have been studied since they include the real spectra of commutative rings [C.N. Delzell, J.J. Madden, J. Algebra 169 (1994) 71], and the prime spectrum of a ring of functions, Spec(C(X)). The existence and importance of this class of non-Hausdorff, normal topological spaces was extremely surprising to us. Further, each of these results is reversible; if the inverse limit is normal, then each space in the spectrum is preceded by one whose bonding map to it is normalizing. By way of contrast, the inverse limit of finite T₀-spaces with separating bonding maps need not be a normal topological space (Example 3.8(a)) and furthermore, if the spaces of the inverse spectrum are normal, then the Hausdorff reflection of the limit must be zero-dimensional (Theorem 3.15).     

Property C. Wallman basis and S-metrizability

In this present paper we prove that every Lindelof space which has a perfect locally connected Hausdorff compactification, has property C. (This latter concept was introduced by R.F. Dickman Jr). We make clear that this class of Lindelöf spaces properly contains the class of paracompact, connected, locally compact and locally connected spaces, as well as the class of those spaces whose topology can be induced by a metric with property S (or S-metrizable spaces). In this fashion, we simultaneously generalize two previous results of Dickman on spaces with property C. The use of Wallman basis with certain connectedness properties turns out to be a very convenient tool in the construction of locally connected compactifications as well as in characterizing S-metrizable spaces.     

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.     

Paths through inverse limits

In Bani?, ?repnjak, Merhar and Milutinovi? (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions fn:X→X² converges to the graph of a continuous single-valued function f:X→X, then the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f. In this paper a more general result is presented in which surjectivity of fn is not required. The result is also generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications, it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.     

Local planarity in one-dimensional continua

In this paper we address a problem posed by W. Lewis at the Second International Conference on Continuum Theory held at BUAP, Puebla, Mexico. Lewis asked for a characterization of local-planarity in inverse limit spaces of finite graphs in terms of the dynamics of the bonding maps. We give some sufficiency conditions and show that points at which our sufficiency conditions do not guarantee the space is locally planar, the problem requires a solution to the harder problem of characterizing planarity in inverse limits of graphs. We also examine the case of an inverse limit generated by a single map, f, on a single graph, G. Assuming that f has finitely many turning points and is non-contracting, we characterize local planarity in terms of the dynamics of f.     

ON THE VITALI-HAHN-SAKS-NIKODYM THEOREM

Abstract

The classical Vitali-Hahn-Saks-Nikodym Theorem [5, Thm. I.4.8] gives a limit criterion for when a sequence of strongly additive vector measures on a σ-field of sets having their range in a Banach space can be expected to be uniformly strongly additive. In [16, Cor. 8], Saeki proved that the limit condition on the sequence of vector measures could be substantially weakened as long as the Banach space in play is “good enough”. Saeki's result was based upon his work on a class of set functions too large to have Rosenthal's Lemma at his disposal. In Section 2, we prove Saeki's result with Rosenthal's Lemma at the basis of our work and then augment our characterization of Banach spaces enjoying Saeki's result in [1] with another natural equivalent condition. In Section 3 we extend Saeki's result to Boolean algebras having the Subsequential Interpolation property.     

Small Valdivia compact spaces

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight?ℵ₁ is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight?ℵ₁ is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.     

Arzelà-Ascoli theorem via the Wallman compactification

In the paper, we recall the Wallman compactification of a Tychonoff space T (denoted by Wall(T)) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between C^b(T), the space of continuous and bounded functions on T , and C(Wall(T)), the space of continuous functions on the Wallman compactification of T. Along the way, we attempt to justify the advantages of the Wallman compactification over other manifestations of the Stone-?ech compactification. The main result of the paper is a new form of the Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along ω-ultrafilters.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Inverse systems and I-favorable spaces

We show that a compact space is I-favorable if, and only if it can be represented as the limit of a σ-complete inverse system of compact metrizable spaces with skeletal bonding maps. We also show that any completely regular I-favorable space can be embedded as a dense subset of the limit of a σ-complete inverse system of separable metrizable spaces with skeletal bonding maps.     

Very I-favorable spaces

We prove that a Hausdorff space X is very I-favorable if and only if X is the almost limit space of a σ-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very I-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.     

Mappings between inverse limits of continua with multivalued bonding functions

We investigate the limit mappings between inverse limits of continua with upper semi-continuous bonding functions. Results are obtained when the coordinate mappings are surjective, one-to-one or homeomorphisms. We construct examples showing the hypothesis of the theorems are essential. Further, we construct an example showing that, unlike for the inverse limits with single valued maps, properties of being monotone, confluent or weakly confluent mappings between factor spaces are not preserved in the inverse limit map.     

On countable-to-one maps

In this paper, it is proved that a space with a point-countable base is an open, countable-to-one image of a metric space, and a quotient, countable-to-one image of a metric space is characterized by a point-countable ℵ₀-weak base. Examples are provided in order to answer negatively questions posed by Gruenhage et al. [G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984) 303-332] and Tanaka [Y. Tanaka, Closed maps and symmetric spaces, Questions Answers Gen. Topology 11 (1993) 215-233].