ALMOST 2-FULLY NORMAL,PAIRWISE PARACOMPACT AND COMPLETE DEVELOPABLE BISPACES

Abstract

Dedicated to Professor Sergio Salbany on the occasion of his 60th birthday.

We introduce and study the notion of an almost 2-fully normal bispace. In particular, we prove that a bispace is quasi-pseudometrizable if and only if it is almost 2-fully normal and pairwise developable. We obtain conditions under which an almost 2-fully normal bispace is subquasi-metrizable and show that the fine quasi-uniformity of any subquasi-metrizable topological space is bicomplete. We prove that every pairwise paracompact bispace (in the sense of Romaguera and Marin, 1988) is almost 2-fully normal and that the finest quasi-uniformity of any 2-Hausdorff pairwise paracompact bispace is bicomplete. We also characterize pairwise paracompactness in terms of a property of σ-Lebesgue type of the finest quasi-uniformity. Finally, we use Salbany's compactification of pairwise Tychonoff bispaces to characterize those bispaces that admit a bicomplete pair development and deduce that an interesting example of R. Fox of a non-quasi-metrizable pairwise stratifiable pairwise developable bispace admits a bicomplete pair development.     

Characterizations of intervals via continuous selections

We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.     

Some remarks concerning C-spaces

We comment on the definition of C-spaces in [D.F. Addis, J.H. Gresham, A class of infinite-dimensional spaces. Part I: Dimension theory and Alexandroff's Problem, Fund. Math. 101 (1978) 195-205] and [W.E. Haver, A covering property for metric spaces, in: Topology Conference at Virginia Polytechnic Institute 1973, in: Lecture Notes in Math., vol. 375, 1974, pp. 108-113]. Furthermore we introduce two types of ‘finite’ C-spaces one of which gives an internal characterization of all spaces having a metrizable compactification satisfying property C. We also introduce a transfinite dimension function for those finite C-spaces. Several questions arise that are related to Alexandrov's problem.     

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.     

Bispaces admitting only bicomplete or only totally bounded quasi-metrics

We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).     

Open retractions onto n-ods

We characterize dendrites which admit an open retraction onto an arbitrary n-od. The paper contains also several corollaries and two problems concerning the subject     

Zero-selectors and GO spaces

We are dealing with Vietoris continuous zero-selectors, i.e., they choose for each non-empty closed set F an isolated point in F. We show that the presence of a continuous zero-selector even on a small class of non-empty closed sets of a space X implies that X is scattered if X is metrizable or non-Archimedean or a P-space. Finally, using continuous zero-selectors, we characterize suborderable spaces which are subspaces of ordinals.     

Ohio completeness and products

In [A.V. Arhangel'ski?, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90], Arhangel'ski? introduced the notion of Ohio completeness and proved it to be a useful concept in his study of remainders of compactifications and generalized metrizability properties. We will investigate the behavior of Ohio completeness with respect to closed subspaces and products. We will prove among other things that if an uncountable product is Ohio complete, then all but countably many factors are compact. As a consequence, Rκ is not Ohio complete, for every uncountable cardinal number κ.     

Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

Some remarks on generalizations of countably compact spaces and Lindelöf spaces

We prove some propositions and present some examples concerning the properties between countably compact and pseudocompact and the properties between Lindelöf and pseudo-Lindelöf.     

A note on monotonically metacompact spaces

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.     

Duality in Function Spaces

In this paper we use Bartle’s technique to study duality between a topological space and a function space. Normally such a duality forms an essential part of Functional Analysis. We introduce several new topologies such as the topology of even convergence T_e, the closed-cocompact topology T_{k ′}, the (strong) local proximal convergence. We explore the topological groups of self-homeomorphisms of a topological space and shed light on the earlier work of Arens, Dieudonné, Di Concilio. We also study the concepts such as evenly equidistant, functionally equicontinuous, due to Bouziad-Troallic and topologically equicontinuous due to Royden. In memory of Professor Enrico Meccariello who made a considerable contribution to this work and who suddenly passed away before his time     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

Decompositions of Borel bimeasurable mappings between complete metric spaces

We prove that every Borel bimeasurable mapping can be decomposed to a σ-discrete family of extended Borel isomorphisms and a mapping with a σ-discrete range. We get a new proof of a result containing the Purves and the Luzin-Novikov theorems as a by-product. Assuming an extra assumption on f, or that Fleissner's axiom (SCω₂) holds, we characterize extended Borel bimeasurable mappings as those extended Borel measurable ones which may be decomposed to countably many extended Borel isomorphisms and a mapping with a σ-discrete range.     

Remainders of rectifiable spaces

We prove a Dichotomy Theorem: for any Hausdorff compactification bG of an arbitrary rectifiable space G, the remainder bG?G is either pseudocompact or Lindelöf. This theorem generalizes a similar theorem on topological groups obtained earlier in A.V. Arhangel'skii (2008) [6], but the proof for rectifiable spaces is considerably more involved than in the case of topological groups. It follows that if a remainder of a rectifiable space G is paracompact or Dieudonné complete, then the remainder is Lindelöf and that G is a p-space. We also present an example showing that the Dichotomy Theorem does not extend to all paratopological groups. Some other results are obtained, and some open questions are formulated.     

Spaces from pieces IV

We are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T₀, T₁, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even they are not really needed).     

Productive local properties of function spaces

We characterize the spaces X for which the space Cp(X) of real valued continuous functions with the topology of pointwise convergence has local properties related to the preservation of countable tightness or the Fréchet property in products. In particular, we use the methods developed to construct an uncountable subset W of the real line such that the product of Cp(W) with any strongly Fréchet space is Fréchet. The example resolves an open question.