Normal covers of infinite products and normality of Σ-products

We give some characterizations for normal covers of infinite products of generalized metric spaces such as M-spaces, Σ-spaces and β-spaces. We prove them simultaneously in terms of β-spaces and perfect maps. Next, we give affirmative answers to two questions concerning the normality of Σ-products, which were raised by the author and Yamazaki, respectively. These results are stated in terms of Σ-products of β-spaces.     

Compact factors in finally compact products of topological spaces

We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors.The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom.     

Linearly ordered covers,normality and paracompactness

It is shown that a regular space is collectionwise normal and countably paracompact if every open cover has an open, order cushioned refinement. A sufficient condition for paracompactness, in terms of certain order locally finite covers, is given, and is applied to the problem of the paracompactness of product spaces.     

Characterizations of normal covers of rectangular products

Stone, Michael and Morita have given various equivalent conditions for normal covers of topological spaces. Here, as an analogue of the classic characterization, we give some characterizations for normal covers of rectangular products in terms of cozero rectangles. Moreover, we apply our characterizations to consider the base-paracompactness of rectangular products.     

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

     

A characterization of the Menger property by means of ultrafilter convergence

We characterize various Menger/Rothberger-related properties, and discuss their behavior with respect to products.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Cosheaves and connectedness in formal topology

The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π₀ together with certain additional structure and properties that constrain π₀ to be the connected components cosheaf. In the inductively generated case, complete spreads (in the sense of Bunge and Funk) corresponding to cosheaves are defined as formal topologies. Maps between the complete spreads are equivalent to homomorphisms between the cosheaves. A cosheaf is the connected components cosheaf for a locally connected formal topology iff its complete spread is a homeomorphism, and in this case it is a terminal cosheaf.A new, geometric proof is given of the topos-theoretic result that a cosheaf is a connected components cosheaf iff it is a “strongly terminal” point of the symmetric topos, in the sense that it is terminal amongst all the generalized points of the symmetric topos. It is conjectured that a study of sites as “formal toposes” would allow such geometric proofs to be incorporated into predicative mathematics.     

Normality of products of monotonically normal spaces with compact spaces

Let S be the class of all spaces, each of which is homeomorphic to a stationary subset of a regular uncountable cardinal (depending on the space). In this paper, we prove the following result: The product X×C of a monotonically normal space X and a compact space C is normal if and only if S×C is normal for each closed subspace S in X belonging to S. As a corollary, we obtain the following result: If the product of a monotonically normal space and a compact space is orthocompact, then it is normal.     

On box products

We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <_ω1?□_α<ω1X_α where each Xα is a compact metric space).     

On nonnormal products and a set of A.H. Stone

A variant of Michael's example is given to the following effect: there is a Lindelöf space M of weight ℵ₁, with all Gδ-sets open, such that M×B(ℵ₁) is nonnormal. This answers a question from [K. Alster, On the class of ω₁-metrizable spaces whose product with every paracompact space is paracompact, Topology Appl. 153 (2006) 2508-2517].     

When does the Haver property imply selective screenability?

We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.     

Selective screenability in topological groups

We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations of Sc(Onbd,O) and Smirnov-Sc(Onbd,O) in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We prove theorems stating conditions under which Sc(Onbd,O) is preserved by products. Among metrizable groups we characterize the countable dimensional ones by a natural game.     

Lebesgue and co-Lebesgue di-uniform texture spaces

The author introduces the notions of Lebesgue di-uniformity and co Lebesgue di-uniformity and investigates the relationship between a Lebesgue quasi uniformity on X and the corresponding Lebesgue di-uniformity on the discrete texture (X,P(X)). Finally a notion of a dual dicovering Lebesgue quasi di-uniform texture space is introduced and several properties are discussed.     

Products of symmetric spaces, and k-networks

As is well known, every product of symmetric spaces need not be symmetric. For symmetric spaces X and Y, in terms of their balls, we give characterizations for the product X×Y to be symmetric under X and Y having certain k-networks, or Y being semi-metric.     

Rectangular products with ordinal factors

Let A and B be subspaces of an ordinal. It is proved that the product A×B is countably paracompact if and only if it is rectangular. Before this main result, we discuss several covering properties of products with one ordinal factor. In particular, for every paracompact space X, it is proved that the product X×A is paracompact if so is A.     

On productively Lindelöf spaces

The class of spaces such that their product with every Lindelöf space is Lindelöf is not well-understood. We prove a number of new results concerning such productively Lindelöf spaces with some extra property, mainly assuming the Continuum Hypothesis.     

Fixed point property on symmetric products

For a metric continuum X, let Fn(X)={A⊂X:A is nonempty and has at most n points}. In this paper we show a continuum X such that F₂(X) has the fixed point property while X does not have it.