Collectionwise Hausdorff Research Articles

A class of nontrivial simple examples of a non-D-space

Abstract Given any regular T 0 {T_{0}} (equivalently, regular T 1 {T_{1}} ) space X, the question of whether X being Lindelöf implies X being a D-space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff T 0 {T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D-space. Also given is a class of handy examples of a second countable hyperconnected T 0 {T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D-space.

Normality Versus Paracompactness in Locally Compact Spaces

AbstractThis note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ω1, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ω1.

The uniform box product

The uniform box product problem is a weakening of the well-known box product problem, which asks whether box products of certain compact spaces are normal or even paracompact. Using uniformities, a new topology on products is defined between the box and Tychonov topologies. This new product, called the uniform box product, is an extension of the sup metric to powers of compact spaces. We investigate a certain non-metrizable compact space whose uniform box product, in ZFC, is normal, countably paracompact, and collectionwise Hausdorff.

PFA(S)[S]: More Mutually Consistent Topological Consequences of PFA and V = L

Abstract Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable L-spaces or compact S-spaces. The model is one of the form PFA(S)[S], where S is a coherent Souslin tree.

Dowker spaces revisited

In 1951, Dowker proved that a space $X$ is countably paracompact and normal if and only if $X \times {\bf I}$ is normal. A normal space $X$ is called a Dowker space if $X \times {\bf I}$ is not normal. The main thrust of this article is to extend this work with regards $\alpha$-normality and $\beta$-normality. Characterizations are given for when the product of a space $X$ and $(\omega + 1)$ is $\alpha$-normal or $\beta$-normal. A new definition, $\alpha$-countably paracompact, illustrates what can be said if the product of $X$ with a compact metric space is $\beta$-normal. Several examples demonstrate that the product of a Dowker space and a compact metric space may or may not be $\alpha$-normal or $\beta$-normal. A collectionwise Hausdorff. Moore space constructed by M. Wage is shown to be $\alpha$-normal but not $\beta$-nornal.

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD ( ℵ 1 ) , fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X × ( ω + 1 ) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD ( ℵ 1 ) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b = ω 1 .

Monotonically countably paracompact, collectionwise Hausdorff spaces and measurable cardinals

We show that, if an MCP (monotonically countably paracompact) space fails to be collectionwise Hausdorff, then there is a measurable cardinal and that, if there are two measurable cardinals, then there is an MCP space that fails to be collectionwise Hausdorff.

Compatible algebraic structures on scattered compacta

It is proved that each hereditarily collectionwise Hausdorff compact scattered space with finite scattered height admits a continuous semilattice operation turning it into a topological semilattice with open principal filters. On the other hand a compactification γ N of a countable discrete space N whose remainder is homeomorphic to [ 0 , ω 1 ] admits no (separately) continuous binary operation turning γ N into an inverse semigroup (semilattice). Also we construct a compactification ψ N of N admitting no separately continuous semilattice operation and such that the remainder ψ N ∖ N is homeomorphic to the one-point compactification of an uncountable discrete space. To show that ψ N admits no continuous semilattice operation we prove that the set of isolated points of a compact scattered topological semilattice X of scattered height 2 is sequentially dense in X. Also we prove that each separable scattered compactum with scattered height 2 is a subspace of a separable compact scattered topological semilattice with open principal filters and scattered height 2. This allows us to construct an example of a separable compact scattered topological semilattice with open principal filters and scattered height 2, which fails to be Fréchet–Urysohn. Also we construct an example of a Fréchet–Urysohn separable non-metrizable compact scattered topological semilattice with open principal filters and scattered height 2.

Kurepa trees and topological non-reflection

A property P of a structure S does not reflect if no substructure of S of smaller cardinality than S has the property. If for a given property P there is such an S of cardinality κ, we say that P does not reflect at κ. We undertake a fine analysis of Kurepa trees which results in defining canonical topological and combinatorial structures associated with the tree which possess a remarkably wide range of nonreflecting properties providing new constructions and solutions of open problems in topology. The most interesting results show that many known properties may not reflect at any fixed singular cardinal of uncountable cofinality. The topological properties we consider vary from normality, collectionwise Hausdorff property to metrizablity and many others. The combinatorial properties are related to stationary reflection.

Uniformization and anti-uniformization properties of ladder systems

Let S denote a stationary subset of limit ordinals of ω1. A ladder system on S is a sequence {Lα : α ∈ S} such that each Lα is an unbounded subset of α of order type ω. A ladder system is uniformizable if for each sequence 〈fα : α ∈ S〉 of functions fα : Lα → ω there is an F : ω1 → ω such that F 1 Lα =∗ fα for each α ∈ S. I.e., for each α ∈ S, {β ∈ Lα : F (β) 6= fα(β)} is finite. We now formulate natural weakenings of uniformizable denoted, for each n ∈ ω, by Pn: A ladder system is said to satisfy Pn if for each f : S → ω there is an F : ω1 → [ω] such that for each α ∈ S (a) F 1 Lα is eventually constant with value sα, and (b) f(α) ∈ sα. Note that P0 is equivalent to the version of uniformizable obtained by considering only sequences of constant functions fα. We will say that a ladder system satisfies P<ω if for each f : S → ω there is an F : ω1 → [ω] satisfying (a) and (b) above. If we drop the requirement that the restrictions F 1 Lα are eventually constant we obtain uniformization properties that we denote Mn and M<ω. E.g., a ladder system is said to satisfy M<ω if for each f : S → ω there is an F : ω1 → [ω] such that for each α ∈ S, f(α) ∈ F (β) for all but finitely many β ∈ Lα. Most of these uniformization properties can be characterized in terms of properties of a certain topological space naturally associated to any ladder system. If L is a ladder system, let XL denote the topology space ω1 × {0} ∪ S × {1} where every point (α, 0) is isolated and for each α ∈ S, a basic neighborhood of (α, 1) consists of {(α, 1)} along with a cofinite subset of Lα×{0}. Such a space is always first countable and locally compact. The stationarity of S implies that it is not collectionwise Hausdorff. Spaces XL have been considered by many to construct examples of normal not collectionwise Hausdorff spaces (see [11] and [2]). It is folklore that a ladder system L satisfies P0 if and only if XL is normal. The property M<ω is characterized by XL being countably metacompact. For this reason, we will say that a ladder system L is countably metacompact in the case that it satisfies M<ω.

Countable paracompactness versus normality in [formula omitted

We will see that: (1) In ZFC, for each subspace X⫅ω 2 1 , the following are equivalent; (a) X is normal, (b) X is countably paracompact and strongly collectionwise Hausdorff, (c) X is expandable. (2) Under a variety of different set-theoretic assumptions (including V = L and PMEA) all countably paracompact subspaces of ω 2 1 are normal. (3) All subspaces of ω 2 1 are collectionwise Hausdorff.

Monotone countable paracompactness

We study a monotone version of countable paracompactness, MCP, and of countable metacompactness, MCM. These properties are common generalizations of countable compactness and stratifiability and are shown to relate closely to the generalized metric g-functions of Hodel: MCM spaces coincide with β-spaces and, for q-spaces (hence first countable spaces) MCP spaces coincide with wN-spaces. A number of obvious questions are answered, for example: there are “monotone Dowker spaces” (monotonically normal spaces that are not MCP); MCP, Moore spaces are metrizable; first countable (or locally compact or separable) MCP spaces are collectionwise Hausdorff (in fact we show that wN-spaces are collectionwise Hausdorff). The extent of an MCP space is shown to be no larger than the density and the stability of MCP and MCM under various topological operations is studied.

On [formula omitted]-discrete, T-finite and tree-type topologies

A regular space is T-finite if and only if it is hereditarily strongly collectionwise Hausdorff and σ-pseudo-closed discrete. Every finer regular topology on such a space is hereditarily ultraparacompact. σ-pseudo-closed discreteness is strictly between σ-closed discreteness and σ-discreteness. It yields ultraparacompactness for regular, strongly collectionwise Hausdorff spaces. Every T-finite, regular topology is finer than a (more or less) canonical topology defined on a tree of height ≤ω. These tree-type topologies (for arbitrary height) are always ultraparacompact and monotonically normal. A space is non-Archimedean and left separated if and only if it is a lob and of tree-type.

Metrizability, monotone normality, and other strong properties in trees

Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFC-independent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.

S‐point finite refinable spaces

A space X is called s‐point finite refinable (ds‐point finite refinable) provided every open cover 𝒰 of X has an open refinement 𝒱 such that, for some (closed discrete) C⫅X,(i) for all nonempty V ∈ 𝒱, V∩C ≠ ∅ and(ii) for all a ∈ C the set (𝒱)a = {V ∈ 𝒱 : a ∈ V} is finite.In this paper we distinguish these spaces, study their basic properties and raise several interesting questions. If λ is an ordinal with cf(λ) = λ &gt; ω and S is a stationary subset of λ then S is not s‐point finite refinable. Countably compact ds‐point finite refinable spaces are compact. A space X is irreducible of order ω if and only if it is ds‐point finite refinable. If X is a strongly collectionwise Hausdorff ds‐point finite refinable space without isolated points then X is irreducible.

On the character of normal non-CWH spaces

It shown that it is consistent (and independent of the Continuum Hypothesis) that every normal space of character at most c + is collectionwise Hausdorff

Failure of normality in the box product of uncountably many real lines

We prove in ZFC that the box product of ω 1 \omega _1 many copies of ω + 1 \omega +1 is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is 2 ω 1 2^{\omega _1} , then these properties also fail in the closed subspace consisting of all functions which assume the value ω \omega on all but countably many indices.

Sequential fans in topology

For an index set I, let S( I) be the sequential fan with I spines, i.e., the topological sum of I copies of the convergent sequence with all nonisolated points identified. The simplicity and the combinatorial nature of this space is what lies behind its occurrences in many seemingly unrelated topological problems. For example, consider the problem which ask us to compute the tightness of the square of S( I). We shall show that this is in fact equivalent to the well-known and more crucial topological question of W. Fleissner which asks whether, in the class of first countable spaces, the property of being collectionwise Hausdorff at certain levels implies the same property at higher levels. Next, we consider Kodama's question whether or not every Σ-product of Lašnev spaces is normal. The sequential fan again enters the scene as we show S( ω 2) × S( ω 2) × ω 1, which can be embedded in a Σ-product of Lašnev spaces as a closed set, can be nonnormal in some model of set theory. On the other hand, we show that the Σ-product of arbitrarily many copies of the slightly smaller fan S( ω 1) is normal.

Reflection and Weakly Collectionwise Hausdorff Spaces

We show that $\square (\theta )$ implies that there is a first countable $< \theta$-collectionwise Hausdorff space that is not weakly $\theta$-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact (supercompact) cardinal to ${\omega _2}$, first countable ${\aleph _1}$-collectionwise Hausdorff spaces are weakly ${\aleph _2}$-collectionwise Hausdorff (weakly collectionwise Hausdorff). In the last section we show that assuming $E_\theta ^\omega$, a certain $\theta$-family of integer-valued functions exists and that in the model obtained by Levy collapsing a supercompact cardinal to ${\omega _2}$, these families do not exist.

Separation and coding

We construct a normal collectionwise Hausdorff space which is not collectionwise normal with respect to copies of [0,1]. We do this by developing a general theory of coding properties into topological spaces. We construct a para-Lindelöf regular space in which para-Lindelöf is coded directly rather than σ \sigma -para-Lindelöf and normal. We construct a normal collectionwise Hausdorff space which is not collectionwise normal in which collectionwise Hausdorff is coded directly rather than obtained as a side-effect to countable approximation. We also show that the Martin’s axiom example of a normal space which is not collectionwise Hausdorff is really just a kind of "dual" of Bing’s space.