Interpolating functions

Let X,Y be sets with quasiproximities X_? and Y_? (where A?B is interpreted as “B is a neighborhood of A”). Let f,g:X→Y be a pair of functions such that whenever CY_?D, then f−1[C]X_?g−1[D]. We show that there is then a function h:X→Y such that whenever CY_?D, then f−1[C]X_?h−1[D], h−1[C]X_?h−1[D] and h−1[C]X_?g−1[D]. Since any function h that satisfies h−1[C]X_?h−1[D] whenever CY_?D, is continuous, many classical “sandwich” or “insertion” theorems are corollaries of this result. The paper is written to emphasize the strong similarities between several concepts

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the posets with auxiliary relations studied in domain theory;

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quasiproximities and their simplification, Urysohn relations; and

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the axioms assumed by Katětov and by Lane to originally show some of these results.

Interpolation results are obtained for continuous posets and Scott domains. We also show that (bi-)topological notions such as normality are captured by these order theoretical ideas.     

Dimension-like functions of the type dim and universality

In [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Math. Stud., vol. 198, Elsevier Science B.V., Amsterdam, 2005, xvi+559 pp.] base dimension-like functions of the type ind, Ind, and dim were defined. The domains of all these functions are classes of pairs (B,X), where B is a base of the space X. Also, positional base dimension-like functions of the type ind were defined whose domains are triples (Q,B,X), where Q is a subset of a space X and B is a base for Q in X. In [D.N. Georgiou, S.D. Iliadis, A.C. Megaritis, Dimension-like functions and universality, Topology Appl. 155 (2008) 2196-2201] and [D.N. Georgiou, S.D. Iliadis, A.C. Megaritis, The universality property for some dimension-like functions, Questions Answers Gen. Topology, in press] some dimension-like functions and positional dimension-like functions of type Ind, respectively are defined. All the above dimension-like functions are studied with respect to the property of universality. Here, we give positional base dimension-like functions of the type dim and prove the property of universality for these functions.     

More generalizations of pseudocompactness

We introduce a covering notion depending on two cardinals, which we call O-[μ,λ]-compactness, and which encompasses both pseudocompactness and many other known generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to O-[ω,ω]-compactness.We provide several characterizations of O-[μ,λ]-compactness, and we discuss its connection with D-pseudocompactness, for D an ultrafilter. The connection turns out to be rather strict when the above notions are considered with respect to products. In passing, we provide some conditions equivalent to D-pseudocompactness.Finally, we show that our methods provide a unified treatment both for O-[μ,λ]-compactness and for [μ,λ]-compactness.     

Monotone versions of δ-normality

According to Mack a space is countably paracompact if and only if its product with [0,1] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular Gδ-set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the subject of this paper. We look at how these properties relate to each other and prove a number of results about them, in particular, we provide a factorization of monotone normality in terms of monotone δ-normality and a weak property that holds in monotonically normal spaces and in first countable Tychonoff spaces. We also discuss the productivity of these properties with a compact metrizable space.     

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.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

On open ultrafilters and maximal points

In this work we expand upon the theory of open ultrafilters in the setting of regular spaces. In [E. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981) 1-45], van Douwen showed that if X is a non-feebly compact Tychonoff space with a countable π-base, then βX has a remote point. We develop a related result for the class of regular spaces which shows that in a non-feebly compact regular space X with a countable π-base, there exists a free open ultrafilter on X that is also a regular filter.Of central importance is a result of Mooney [D.D. Mooney, H-bounded sets, Topology Proc. 18 (1993) 195-207] that characterizes open ultrafilters as open filters that are saturated and disjoint-prime. Smirnov [J.M. Smirnov, Some relations on the theory of dimensions, Mat. Sb. 29 (1951) 157-172] showed that maximal completely regular filters are disjoint prime, from which it was concluded that βX is a perfect extension for a Tychonoff space X. We extend this result, and other results of Skljarenko [E.G. Skljarenko, Some questions in the theory of bicompactifications, Amer. Math. Soc. Transl. Ser. 2 58 (1966) 216-266], by showing that a maximal regular filter on any Hausdorff space is disjoint prime.Open ultrafilters are integral to the study of maximal points and lower topologies in the partial order of Hausdorff topologies on a fixed set. We show that a maximal point in a Hausdorff space cannot have a neighborhood base of feebly compact neighborhoods. One corollary is that no locally countably compact Hausdorff topology is a lower topology, which was shown previously under the additional assumption of countable tightness by Alas and Wilson [O. Alas, R. Wilson, Which topologies can have immediate successors in the lattice of T₁-topologies? Appl. Gen. Topol. 5 (2004) 231-242]. Another is that a maximal point in a feebly compact space is not a regular point. This generalizes results of both Carlson [N. Carlson, Lower upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007) 619-624] and Costantini [C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008) 187-225].     

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.     

D-spaces, aD-spaces and finite unions

In this paper, we prove that if a space X is the union of a finite family of strong Σ-spaces, then X is a D-space. This gives a positive answer to a question posed by Arhangel'skii in [A.V. Arhangel'skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (2004) 2163-2170]. We also obtain results on aD-spaces and finite unions. These results improve the correspond results in [A.V. Arhangel'skii, R.Z. Buzyakova, Addition theorems and D-spaces, Comment. Math. Univ. Carolin. 43 (2002) 653-663] and [Liang-Xue Peng, The D-property of some Lindelöf spaces and related conclusions, Topology Appl. 154 (2007) 469-475].     

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.     

Selection principles in hyperspaces with generalized Vietoris topologies

We continue investigations of ?ech closure spaces and their hyperspaces started in [M. Mrševi?, M. Jeli?, Selection principles and hyperspace topologies in closure spaces, J. Korean Math. Soc. 43 (2006) 1099-1114] and [M. Mrševi?, M. Jeli?, Selection principles, γ-sets and αi-properties in ?ech closure spaces, Topology Appl., in press], focusing on generalized upper and lower Vietoris topologies.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

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.     

Approximations of relations by continuous functions

Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

Domination by metric spaces

Following the definition of domination of a topological space X by a metric space M introduced by Cascales, Orihuela and Tkachuk (2011) in [3], we define a topological cardinal invariant called the metric domination index of a topological space X   as minimum of the set {w(M):M is a metric space that dominates X}$\{w(M):M\text{is a metric space that dominates}X\}$. This invariant quantifies or measures the concept of M-domination of Cascales et al. (2011) [3]. We prove (in ZFC) that if K   is a compact space such that _Cp(K)$C_p(K)$ is strongly dominated by a second countable space then K is countable. This answers a question by the authors of Cascales et al. (2011) [3].     

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X×[0,1] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27-36].     

Cover quasi-uniformities in frames

Quasi-uniformities (not necessarily symmetric uniformities) are usually studied via entourages (special neighbourhoods of the diagonal in X×X) where one can simply forget about the symmetry requirement. This has been done successfully in the point-free context as well, but there is a demand for a covering approach, a.o. because the point-free representation of the square X×X is not without difficulties. Based on the (spatial) ideas from Gantner and Steinlage (1972) [9], a cover type quasi-uniformity was developed in Frith (1987) [6] and other papers using biframes, the point-free variant of bitopologies. In this paper we show that this can be avoided and present a cover type quasi-uniformity structure enriching that of frame directly.