A classification of CO spaces which are continuous images of compact ordered spaces

A topological space X is called a CO space, if every closed subset of X is homeomorphic to some clopen subset of X. Every ordinal with its order topology is a CO space. This work gives a complete classification of CO spaces which are continuous images of compact ordered spaces.     

The structure of Valdivia compact lines

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed ℵ₁, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.     

Preorderable topologies and order-representability of topological spaces

We furnish characterizations of topologies that coincide with the lower topology or with the order topology of some total preorder defined on a set. Leaning on these characterizations we introduce some applications to the study of continuous and semicontinuous order-representability properties of topological spaces.     

When is a Volterra space Baire?

In this paper, we study the problem when a Volterra space is Baire. It is shown that every stratifiable Volterra space is Baire. This answers affirmatively a question of Gruenhage and Lutzer in [G. Gruenhage, D. Lutzer, Baire and Volterra spaces, Proc. Amer. Math. Soc. 128 (2000) 3115-3124]. Further, it is established that a locally convex topological vector space is Volterra if and only if it is Baire; and the weak topology of a topological vector space fails to be Baire if the dual of the space contains an infinite linearly independent pointwise bounded subset.     

Game approach to universally Kuratowski-Ulam spaces

We consider a version of the open-open game, indicating its connections with universally Kuratowski-Ulam spaces. From [P. Daniels, K. Kunen, H. Zhou, On the open-open game, Fund. Math. 145 (3) (1994) 205-220] and [D. Fremlin, T. Natkaniec, I. Rec?aw, Universally Kuratowski-Ulam spaces, Fund. Math. 165 (3) (2000) 239-247] topological arguments are extracted to show that: Every I-favorable space is universally Kuratowski-Ulam, Theorem 8; If a compact space Y is I-favorable, then the hyperspaceexp(Y)with the Vietoris topology is I-favorable, and hence universally Kuratowski-Ulam, Theorems 6 and 9. Notions of uK-U and uK-U^∗ spaces are compared.     

On the compact open and finest splitting topologies

In [M.H. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145] it is shown that in the set C(Nω,N) of all continuous maps of Nω into N, where N is an infinitely countable discrete topological space, the compact-open topology is not the finest splitting topology. Since Nω is consonant (see [S. Dolecki, G.H. Greco, A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? Trans. Amer. Math. Soc. 347 (1995) 2869-2884]) the Isbell topology on C(Nω,N) also is not the finest splitting topology. This result is generalized in the present paper proving that it is true also for spaces having the so-called Specific Extension Property. The following spaces have the Specific Extension Property: (a) infinitely countable free unions of non-empty spaces, (b) non-compact Lindelöf zero-dimensional spaces, and (c) metric locally convex linear spaces. In particular, we prove that on the set of all real-valued functions on the (separable infinite dimensional) Hilbert space the compact-open topology does not coincide with the finest splitting topology.     

Range universal spaces

We characterize those topological spaces Y for which the Isbell and finest splitting topologies on the set C(X,Y) of all continuous functions from X into Y coincide for all topological spaces X. We also consider the same question for the coincidence of the restriction of the finest splitting topology on the upper semicontinuous set-valued functions to C(X,Y) and the finest splitting topology on C(X,Y). In the first case, the spaces in question are, after identifying points that are in each others closures, subsets of the two point Sierpiński space, which gives a converse and generalization of a result of S. Dolecki, G.H. Greco, and A. Lechicki. In the second case, the spaces in question are, after identifying points that are in each others closures, order bases for bounded complete continuous DCPOs with the Scott topology.     

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.     

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.     

Embedding in R-closed spaces

Some problems in the theory of R-closed spaces are solved by showing that every regular space can be embedded in a minimal regular space and there is an R-closed space with no coarser minimal regular topology. A class of spaces is found so that when fed into the Jone's machinery for producing non-Tychonoff, regular spaces, the output is non-tychonoff R-closed and minimal regular spaces. Also, an example of a strongly minimal regular space that is not locally R-closed is given.     

On the compact-open and admissible topologies

It is known (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, Function space topologies, in: Open Problems in Topology 2, Elsevier, 2007, pp. 15-23]) that 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 (which in general is not admissible). The following, interesting in our opinion, problem is arised: when a given splitting topology (for example, the compact-open topology, the Isbell topology, and the greatest splitting topology) is the intersection of k admissible topologies, where k is a finite number. Of course, in this case this splitting topology will be the greatest splitting.In the case, where a given splitting topology is admissible the above number k is equal to one. For example, if Y is a locally compact Hausdorff space, then k=1 for the compact-open topology (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]). Also, if Y is a corecompact space, then k=1 for the Isbell topology (see [P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191-211; F. Schwarz, S. Weck, Scott topology, Isbell topology, and continuous convergence, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251-271]).In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] a non-locally compact completely regular space Y is constructed such that the compact-open topology on C(Y,S), where S is the Sierpinski space, coincides with the greatest splitting topology (which is not admissible). This fact is proved by the construction of two admissible topologies on C(Y,S) whose intersection is the compact-open topology, that is k=2.In the present paper improving the method of [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] we construct some other non-locally compact spaces Y such that the compact-open topology on C(Y,S) is the intersection of two admissible topologies. Also, we give some concrete problems concerning the above arised general problem.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.     

Upper semifinite hyperspaces as unifying tools in normal Hausdorff topology

In this paper we use the upper semifinite topology in hyperspaces to get results in normal Hausdorff topology. The advantage of this point of view is that the upper semifinite topology, although highly non-Hausdorff, is very easy to handle. By this way we treat different topics and relate topological properties on spaces with some topological properties in hyperspaces. This hyperspace is, of course, determined by the base space. We prove here some reciprocals which are not true for the usual Vietoris topology. We also point out that this framework is a very adequate one to construct the ?ech-Stone compactification of a normal space. We also describe compactness in terms of the second countability axiom and of the fixed point property. As a summary we relate non-Hausdorff topology with some facts in the core of normal Hausdorff topology. In some sense, we reinforce the unity of the subject.     

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.     

Mappings onto metric spaces

We give characterizations of perfect images and open and compact images of spaces that can be mapped onto metrizable spaces by a mapping with fibers having a given property $\text{P}$. We use these characterizations to obtain conditions which imply that such images can be mapped onto a metric space by a mapping with fibers satisfying $\text{P}$. Such a treatment includes the investigation of spaces with a weaker metric topology [2, Ch. 5].     

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.     

Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces

The concept of lower semicontinuity is extended to functions mapping into partially ordered spaces. A study is made of spaces of such lower semicontinuous functions under the epi-topology. These spaces are subspaces of hyperspaces with the Fell topology. The closure of such a function space in the hyperspace is characterized for certain spaces. A continuous selection theorem is established, showing that most such function spaces are not ech-complete.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.