Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution RK(X) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then RK(X) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X×P. Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X^′, such that X is strong shape dominated by X^′ and the Cartesian product X^′×P is a direct product in Sh(Top), then also X×P is a direct product in the shape category Sh(Top).     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron II

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In a subsequent paper, published in 2007, the author proved that R(X,K) is a covariant functor in the first variable. In the present paper it is proved that R(X,K) is a covariant functor also in the second variable.     

Products of Compacta with Plyhedra and Topological Spaces in the Shape Category

In the literature there exist examples of separable metric spaces X,Y whose Cartesian product X × Y is not a product in the shape category Sh(Top). It is an open question whether, for X a compact Hausdorff space, X × Y is a product in Sh(Top), for every topological spaces Y. The main result of the paper asserts that the answer is positive provided X × P is a product in Sh(Top), for every polyhedron P.     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

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.     

Canonical subbase-compactness of topological products

For topological products the concept of canonical subbase-compactness is introduced, and the question analyzed under what conditions such products are canonically subbase-compact in ZF-set theory.Results: (1) Products of finite spaces are canonically subbase-compact iff AC(fin), the axiom of choice for finite sets, holds.(2) Products of n-element spaces are canonically subbase-compact iff AC(<n), the axiom of choice for sets with less than n elements, holds.(3) Products of compact spaces are canonically subbase-compact iff AC, the axiom of choice, holds.(4) All powers XI of a compact space X are canonically subbase compact iff X is a Loeb-space.These results imply that in ZF the implications

     

Some remarks on a game of Telgársky

In [3] R. Telgársky (1975) asked: does the first player have a winning strategy in the game G(F,X×X) if the first player has a winning strategy in the game G(F,X)? I give a positive answer to this question and prove that this result is also true for spaces where the first player has a winning strategy in game G(K,X) where K=1, F, C, for σC if X is P-space and for DC if X is collectionwise-normal space. The last result is related to the Telgársky's (1983) conjecture discussed in [1]. These results are not true for infinite product of spaces.     

PRODUCTS IN TOPOLOGY

Abstract

Examples are provided which demonstrate that in many cases topological products do not behave as they should. A new product for topological spaces is defined in a natural way by means of interior covers. In general this is no longer a topological space but can be interpreted as categorical product in a category larger than Top. For compact spaces the new product coincides with the old. There is a converse: For symmetric topological spaces X the following conditions are equivalent: (1) X is compact; (2) for each cardinal k the old and the new product X^k coincide; (3) for each compact Hausdorff space Y the old and the new product X x Y coincide. The new product preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelöf property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete.     

Essential extensions of Hausdorff spaces

In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).Dedicated to my friend Nico Pumplün on his sixtieth birthday     

Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums (=coproducts) and products. However, it may happen that for families (Xi)_i∈I and (Yi)_i∈I with the property that each Xi is homeomorphic to the corresponding Yi neither their sums _⊕i∈IXi and _⊕i∈IYi nor their products _∏i∈IXi and _∏i∈IYi are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.     

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.     

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.     

The Topology of Limits of Direct Systems Induced by Maps

Let Z, H be spaces. In previous work, we introduced the direct system X induced by the set of maps between the spaces Z and H. Now we will consider the case that X is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim X will be T₁, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties.     

On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

2-Categorical aspects of strong shape

For every pair (C,K) of groupoid enriched categories, we define two related categories S(C,K) and SS(C,K). In the case (C,K)=(TOP,ANR) they are isomorphic, respectively, to the classical shape category Sh(TOP) of Mardeši? and Segal and to the strong shape category of topological spaces SSh(1)(TOP) of height 1, defined by Lisicá and Mardeši?. Moreover, for (C,K)=(CM,ANR), where CM is the category of compact metric spaces, there is an isomorphism SSh(CM)≅SS(CM,ANR). A new characterization of topological strong shape equivalences is also given.     

More on remainders close to metrizable spaces

This article is a natural continuation of [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90]. As in [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90], we consider the following general question: when does a Tychonoff space X have a Hausdorff compactification with a remainder belonging to a given class of spaces? A famous classical result in this direction is the well known theorem of M. Henriksen and J. Isbell [M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958) 83-106].It is shown that if a non-locally compact topological group G has a compactification bG such that the remainder Y=bG?G has a Gδ-diagonal, then both G and Y are separable and metrizable spaces (Theorem 5). Several corollaries are derived from this result, in particular, this one: If a compact Hausdorff space X is first countable at least at one point, and X can be represented as the union of two complementary dense subspaces Y and Z, each of which is homeomorphic to a topological group (not necessarily the same), then X is separable and metrizable (Theorem 12). It is observed that Theorem 5 does not extend to arbitrary paratopological groups. We also establish that if a topological group G has a remainder with a point-countable base, then either G is locally compact, or G is separable and metrizable.     

Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⇒β(X)τK for all paracompact spaces X. Here are the main results of the paper:

Theorem 0.1. If{Ks}_s∈Sis a family of pointed quasi-finite complexes, then their wedge_?s∈SKsis quasi-finite.