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.     

The smallest basically disconnected preimage of a space

We show that for each space X, there exists a smallest basically disconnected perfect irreducible preimage ΛX. A corollary of the existence of ΛX is that each locally compact and basically disconnected space X has a smallest basically disconnected compactification.     

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.     

Equivariant strong shape

In this paper we propose a construction of the equivariant strong shape for compact metrizable G-spaces using an equivariant version of so-called cotelescopes and the concept of a fibrant G-space.     

Universal space in the Cartesian product of Peano curves without free arcs

We prove that there is the universal space for the class of n-dimensional separable metric spaces in the Cartesian product K₁×?×Kn+1 of Peano curves without free arcs. It is also shown that the set of embeddings of any n-dimensional separable metric space X into this universal space is a residual set in C(X,K₁×?×Kn+1). Other properties of product of Peano curves without free arcs are also proved.     

Strong shape theory and resolutions

It is shown that the strong shape theory of compact metrizable spaces extends to a theory for all topological spaces. The extension resembles the inverse systems approach to shape theory of Marde?i? and Segal. Fundamental roles are played by the Steenrod homotopy theory of Edwards and Hastings and the theory of ANR-resolutions due to Marde?i?.     

Embedding Tn-like continua in Euclidean space

Many authors have been concerned with embedding ∏-like continua in Rⁿ where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rⁿ up to shape. In this paper we prove two main theorems. Theorem: If n ? 2 and X is Tⁿ-like, then X embeds in R²ⁿ. This result was conjectured by McCord for the case H¹(X) finitely generated and proved by McCord for the case that H¹(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tⁿ-like, then X embeds in Rⁿ⁺² up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rⁿ⁺². Any Tⁿ-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ? k ? n.     

On a question of Mauldin and Ulam concerning homeomorphisms

In this paper we solve a question of Mauldin and Ulam about transformations preserving homeomorphic pairs.     

Monomorphisms and epimorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. We give characterizations of monomorphisms (respectively, epimorphisms) in pro-category pro-C, provided C has direct sums (respectively, pushouts).Let E(C) (respectively, M(C)) be the subcategory of C whose morphisms are epimorphisms (respectively, monomorphisms) of C. We give conditions in some categories C for an object X of pro-C to be isomorphic to an object of pro-E(C) (respectively, pro-M(C)).A related class of objects of pro-C consists of X such that there is an epimorphism X→P∈Ob(C) (respectively, a monomorphism P∈Ob(C)→X). Characterizing those objects involves conditions analogous (respectively, dual) to the Mittag-Leffler property. One should expect that the object belonging to both classes ought to be stable. It is so in the case of pro-groups. The natural environment to discuss those questions are balanced categories with epimorphic images. The last part of the paper deals with that question in pro-homotopy.     

Comparing weak versions of separability

Our aim is to investigate spaces with σ-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [22], Hutchison [13] and the authors of [7].     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².     

The rational line partitions every self-dense metrisable space

In the present note we shall prove that a metrisable space can be partitioned by the rational line if that space is self-dense. This gives an affirmative answer to a question raised by Rankston-McGovern [1].     

Proper hereditary shape equivalences preserve property C

In a recent paper [6], van Mill and Mogilski prove that a proper hereditary shape equivalence preserves property C, if its domain is σ-compact. In this note, the same result is established without the hypothesis of σ-compactness.     

Every scattered space is subcompact

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every ?ech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω  -monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense _Gδ$G_{\delta }$-subsets of Cantor cubes are subcompact.     

On a local property of products of Lašnev spaces

Let p denote a free ultrafilter on the natural numbers N. It is shown that N ? {p} cannot be embedded in any countable product of La?nev spaces.     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

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