Complete systems, elementary submodels and the tightness of upper hyperspaces

We introduce and study some completeness properties for systems of open coverings of a given topological space. A Hausdorff space admitting a system of cardinality κ satisfying one of these properties is of type Gκ. Hence, we define several new variants of the ?ech number and use elementary submodels to determine further results. We introduce M-hulls and M-networks, for M elementary submodel. As an application, we give estimates for both the tightness and the Lindelöf number of a generic upper hyperspace. Two recent results of Costantini, Holá and Vitolo on the tightness of co-compact hyperspaces follow from ours as corollaries.     

Lower and upper topologies in the Hausdorff partial order on a fixed set

In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ?σ in which there is no Hausdorff topology μ satisfying σ?μ?τ. τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies.     

Compact Hausdorff fuzzy topological spaces are topological

Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces.     

Vector topologies and the representation of spaces of random variables

Most of the spaces of random variables are non-Hausdorff linear topological spaces, whereas a number of useful results of the theory of vector topologies require the Hausdorff separation axiom for the underlying spaces. Therefore we review and complete the machinery of non-Hausdorff linear topological spaces and the techniques for their representation with suitable Hausdorff linear topological spaces, and then we work out some examples, which emphasize the ideas involved in a consistent application of the topological point of view. The final part of the paper is devoted to the discussion of the pathology, which arises when we try to treat the special case of almost everywhere convergence in the same framework.     

Hereditary normality-type properties of hyperspaces

Let X be a Hausdorff topological space and exp(X) be the space of all (nonempty) closed subsets of a space X with the Vietoris topology. We consider hereditary normality-type properties of exp(X). In particular, we prove that if exp(X) is hereditarily D-normal, then X is a metrizable compact space.     

Homotopical properties of upper semifinite hyperspaces of compacta

In this paper we study homotopical properties of a special neighborhood system, which is denoted by {Uε}_?>0, for the canonical embedding of a compact metric space in its upper semifinite hyperspace to get results in the shape theory for compacta. We also point out that there are spaces with the shape of finite discrete spaces and having not the homotopy type of any T₁-space     

Normal Vietoris Implies Compactness: A Short Proof

One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.     

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].     

Expanding topological space,study and applications

In this paper, we introduce the notion of expanding topological space. We define the topological expansion of a topological space via local multi-homeomorphism over coproduct topology, and we prove that the coproduct family associated to any fractal family of topological spaces is expanding. In particular, we prove that the more a topological space expands, the finer the topology of its indexed states is. Using multi-homeomorphisms over associated coproduct topological spaces, we define a locally expandable topological space and we prove that a locally expandable topological space has a topological expansion. Specifically, we prove that the fractal manifold is locally expandable and has a topological expansion.     

Metrizability of paratopological (semitopological) groups

We investigate some generalized metric space properties on paratopological (semitopological) groups and prove that a paratopological group that is quasi-metrizable by a left continuous, left-invariant quasi-metric is a topological group and give a negative answer to Ravsky?s question (Ravsky, 2001 [18, Question 3.1]). It is also shown that an uncountable paratopological group that is a closed image of a separable, locally compact metric space is a topological group. Finally, we discuss Hausdorff compactification of paratopological (semitopological) groups, give an affirmative answer to Lin and Shen?s question (Lin and Shen, 2011 [14, Question 6.9]) and improve an Arhangel?skii and Choban?s theorem. Some questions are posed.     

Aspects of the Qualitative Theory of Suspended Foliations

We proved that the set of suspended foliations of a compact manifold M is open in the space of foliations with C r -topology. Relation between sets of suspended foliations with Hausdorff and non-Hausdorff graphs are investigated from topological and set-theoretical points of view. Continuum set of pairwise non-isomorphic C X -diffeomorphisms of 1-dimensional manifold with special properties is constructed and used essentially.     

On the greatest splitting topology

It is well known that (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, in: Elliott Pearl (Ed.), Function Space Topologies, Open Problems in Topology, vol. 2, Elsevier, 2007, pp. 15-22]) the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology. However, this intersection maybe not admissible. In the case, where Y is a locally compact Hausdorff space the compact-open topology on the set C(Y,Z) is splitting and admissible (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]), which means that the intersection of all admissible topologies on C(Y,Z) is admissible. In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] an example of a non-locally compact Hausdorff space Y is given having the same property for the case, where Z=[0,1], that is on the set C(Y,[0,1]) the compact-open topology is splitting and admissible. This space Y is the set [0,1] with a topology τ, whose semi-regular reduction coincides with the usual topology on [0,1]. Also, in [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31, Theorem 5.3] another example of a non-locally compact space Y is given such that the compact-open topology on the set C(Y,[0,1]) is distinct from the greatest splitting topology.In this paper first we construct non-locally compact Hausdorff spaces Y such that the intersection of all admissible topologies on the set C(Y,Z), where Z is an arbitrary regular space, is admissible. Furthermore, for a Hausdorff splitting topology t on C(Y,Z) we find sufficient conditions in order that t to be distinct from the greatest splitting topology. Using this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on C(Y,Z), where Z is a Hausdorff space, is distinct from the greatest splitting topology. Finally, we give some open problems.     

An extension of the dual complexity space and an application to Computer Science

In 1999, Romaguera and Schellekens introduced the theory of dual complexity spaces as a part of the development of a mathematical (topological) foundation for the complexity analysis of programs and algorithms [S. Romaguera, M.P. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311-322]. In this work we extend the theory of dual complexity spaces to the case that the complexity functions are valued on an ordered normed monoid. We show that the complexity space of an ordered normed monoid inherits the ordered normed structure. Moreover, the order structure allows us to prove some topological and quasi-metric properties of the new dual complexity spaces. In particular, we show that these complexity spaces are, under certain conditions, Hausdorff and satisfy a kind of completeness. Finally, we develop a connection of our new approach with Interval Analysis.     

Generalized Alexandroff-Urysohn squares and a characterization of the fixed point property

Given a Hausdorff continuum X, we introduce a topology on X×X that yields a Hausdorff continuum. We call the resulting space the Alexandroff-Urysohn square of X and prove that X has the fixed point property if and only if the Alexandroff-Urysohn square of X has the fixed point property.     

Applications of k-covers II

We continue the study of applications of k-covers to some topological constructions, mostly to function spaces and hyperspaces.     

The Baire property in hit-and-miss hypertopologies

Recently, some techniques have been developed for the study of the Baire property in hyperspaces. These techniques have been applied to solve a long-standing open problem of McCoy in 1975 and a recent open problem of Zsilinszky. In this paper, we extend and apply these techniques further to investigate the Baire property of hyperspaces equipped with the general hit-and-miss topology.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

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).     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

Fell topology on the hyperspace of a non-Hausdorff space

Fell topology is very widely used today, even in metric spaces; but J. Fell introduced it in a non-Hausdorff context in the connection with the theory of C ^*-algebras. In spite of this, it has been studied only on the hyperspace of a Hausdorff space, except for the first results due to Fell himself. The present paper aims to fill this gap, in particular extending some results of H. Poppe and of G. Beer to the general case. Project 10251002 supported by National Natural Science Foundation of China.