Dowker Space Research Articles

Semi-proximal spaces and normality

We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of (ω+1)×ω1. In contrast, we show that every normal subspace of a finite power of ω1 is semi-proximal.

A new small Dowker space

A new small Dowker space

Locally compact, ω1-compact spaces

An ω1-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ω1-compact space is σ-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, ω1-compact space of cardinality ℵ1 is σ-countably compact. Whether ℵ1 can be replaced with ℵ2 is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, ω1-compact space is σ-countably compact.As a result, it is also ZFC-independent whether there is a locally compact, ω1-compact Dowker space of cardinality ℵ1, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space ω1.Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as “MM(S)[S]”. Most of the work is done by the P-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality ℵ1, as it is in several theorems.

Locally compact, monotonically normal Dowker space in ZF+AD

We construct a locally compact, monotonically normal, screenable, hereditarily paracompact space Y on P×ω in ZFC. The space Y is also a D-space in ZFC. The space Y is shown to be a locally compact, monotonically normal Dowker space and is not a D-space under ZF+AD and other models of set-theory.

Paracompact in ZFC; CWN screenable Dowker in ZF+AD

Using a previously unexplored technique we construct a (transparently) hereditarily normal, collectionwise normal, screenable space Z on P×ω in ZFC. With more effort, the space Z is shown to be hereditarily paracompact and is hereditarily a D-space in ZFC. However, Z is easily shown to be a collectionwise normal, screenable Dowker space and is not a D-space in ZF+AD. We also observe that Z is a monotonically normal Dowker space in ZF+AD. It is known that such do not exist in ZFC. If one wants to avoid large cardinals there is a model due to S. Shelah [16], equiconsistent with ZF, where Z is a CWN screenable Dowker space.

Comment on ‘Wedges, cones, cosmic strings and their vacuum energy’

A recent paper (Fulling et al 2012 J. Phys. A: Math. Theor. 45 374018) is extended by investigating the behavior of the regularized quantum scalar stress tensor near the axes of cones and their covering manifold, the Dowker space. A cone is parametrized by its angle θ1, where θ1 = 2π for flat space. We find that the tensor components have singularities of the type rγ, but the generic leading γ equals , which is negative if and only if θ1 > 2π, and is a positive integer if . Thus the functions are analytic in those cases that can be solved by the method of images starting from flat space, and they are not divergent in the cases that interpolate between those. As a wedge of angle α can be solved by images starting from a cone of angle 2α, a divergent stress can arise in a wedge with π < α ⩽ 2π but not in a smaller one.

Wedges, cones, cosmic strings and their vacuum energy

One of J Stuart Dowker’s most significant achievements has been to observe that the theory of diffraction by wedges developed a century ago by Sommerfeld and others provided the key to solving two problems of great interest in general-relativistic quantum field theory during the last quarter of the 20th century: the vacuum energy associated with an infinitely thin, straight cosmic string, and (after an interchange of time with a space coordinate) the apparent vacuum energy of empty space as viewed by an accelerating observer. In a sense the string problem is more elementary than the wedge, since Sommerfeld’s technique was to relate the wedge problem to that of a conical manifold by the method of images. Indeed, Minkowski space, as well as all cone and wedge problems, are related by images to an infinitely sheeted master manifold, which we call Dowker space. We review the research in this area and exhibit in detail the vacuum expectation values of the energy density and pressure of a scalar field in Dowker space and the cone and wedge spaces that result from it. We point out that the (vanishing) vacuum energy of Minkowski space results, from the point of view of Dowker space, from the quantization of angular modes, in precisely the way that the Casimir energy of a toroidal closed universe results from the quantization of Fourier modes; we hope that this understanding dispels any lingering doubts about the reality of cosmological vacuum energy.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

Borel extensions of Baire measures in ZFC

We prove: (1) Every Baire measure on the Kojman-Shelah Dowker space [10] admits a Borel extension. (2) If the continuum is not a real-valued measurable cardinal then every Baire measure on the M. E. Rudin Dowker space [16] admits a Borel extension. Consequently, Balogh’s space [3] remains as the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

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.

Normality in products with a countable factor

The class of normal spaces that have normal product with every countable space is considered. A countably compact normal space X and a countable Y such that X × Y is not normal is constructed assuming CH. Also, ⋄ is used to construct a perfectly normal countably compact X and a countable Y such that X × Y is not normal. The question whether a Dowker space can have normal product with itself is considered. It is shown that if X is Dowker and contains any countable non-discrete subspace, then X 2 is not normal. It follows that a product of a Dowker space and a countable space is normal if and only if the countable space is discrete. If X is Rudin's ZFC Dowker space, then X 2 is normal. An example of a Dowker space of cardinality ℵ 2 with normal square is constructed assuming ⋄ ω 2 ∗ .

A study of remainders of topological groups

Some duality theorems relating properties of topological groups to properties of their remainders are established. It is shown that no Dowker space can be a remainder of a topological group. Perfect normality of a remainder of a topological group is consi

Rudin's Dowker space in the extension with a Suslin tree

We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle refining property for forcing notions, which modifies the one for partitions due to Paul B. Larson and Stevo Todorčević and is stronger

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X × [ 0 , 1 ] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27–36].

Rudin's Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F 0 , F 1 of a normal T 1 -space X admits a star-finite open cover U of X such that, for every U ∈ U , either U ¯ ∩ F 0 = ∅ or U ¯ ∩ F 1 = ∅ holds. We define a T 1 -space X to be strongly base-normal if there is a base B for X with | B | = w ( X ) satisfying that every pair of disjoint closed subsets F 0 , F 1 of X admits a star-finite cover B ′ of X by members of B such that, for every B ∈ B ′ , either B ¯ ∩ F 0 = ∅ or B ¯ ∩ F 1 = ∅ holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.

Results about κ-normality

A regular topological space is called κ- normal if any two disjoint κ-closed subsets in it are separated. In this paper we present some results about κ-normality. In Section 1, the technique of adding isolated points to topological spaces has been used to construct three counterexamples. The first one shows the following statements: (1) A product of two linearly ordered topological spaces need not be κ- normal even if one of the factors is compact. (2) A product of two normal countably compact topological spaces need not be κ- normal. (3) A product of a normal topological space with a compact Hausdorff topological space need not be κ- normal. The second one presents a mad family R⊂[ω] ω such that the Mrówka space Ψ( R) is not κ-normal. The third one will show that a scattered locally compact countably compact topological space need not be κ- normal. In Section 2 we have proved the following κ-normal version of Stone's theorem: If X is κ- normal and countably compact and Y is metrizable, then X× Y is κ- normal. For a κ-normal version of Dowker's theorem, we have been able to prove one direction which is the following statement: If X is not κ- countably metacompact, then X× I is not κ- normal. We use I for the closed unit interval [0,1] with the usual topology. The converse is still unsettled. We will show that if X is a Dowker space, then the Alexandroff Duplicate space A( X) of X is a Dowker space with the property that A( X)× I is not κ- normal. Section 3 has been devoted to the notion of local κ-normality. It will be shown that not every locally κ-normal topological space is κ-normal, even if the space satisfies other topological properties such as locally compactness, metacompactness, or countable compactness.

Dowker spaces and paracompactness questions

We construct, in ZFC, a hereditarily collectionwise normal, hereditarily metaLindelöf, hereditarily realcompact Dowker space. This answers a question of R. Hodel (also asked by S. Watson and D. Burke) and another question of M.E. Rudin.

A ZFC Dowker space in ℵ_{𝜔+1}: An application of pcf theory to topology

The existence of a Dowker space of cardinality ℵ ω + 1 \aleph _{\omega +1} and weight ℵ ω + 1 \aleph _{\omega +1} is proved in ZFC using pcf theory.

Killing normality with a Cohen real

We describe a machine that inputs a normal space X and outputs a normal superspace T such that T becomes nonnormal after adding one Cohen real if and only if X is a Dowker space. A similar construction applied to Rudin's box product Dowker space yields a collectionwise normal space that becomes nonnormal after the addition of one Cohen real.

A small Dowker space in ZFC

We construct a hereditarily normal topological space whose product with the unit interval is not normal. The space is σ \sigma -relatively discrete and has cardinality of the continuum c \mathfrak {c} .

Complementary forms of [α, β]-compact

For infinite cardinals κ λ, let X ϵ C( κ, λ) iff for each open cover U of X of cardinality less than λ, X - U is finally κ-compact (i.e., [κ, β]-compact for all β) for some U ϵ U . The T 3 spaces in C(ω, ω) comprise the NAC spaces of W. Fleissner et al. Their cofinality, cf( X), extends naturally to C(κ, λ) and, combined with a new cardinal invariant, cfc( X), completely determines C(κ, λ), i.e., X ϵ C(κ, λ) iff κ ⩾ cfc( X) and λ ⩽ cf( X). The subclasses C(κ, ω) of cofinally κ-compact and C(κ, κ) of κ-cocompact spaces are more interesting and give meaning to cf( X) and cfc( X). They relate to Dowker space constructions of M.E. Rudin and others, e.g. the inclusion C ( κ, κ) ⊃ C( κ, ω) is made proper by some of these non-ZFC spaces. Section 2 gives equivalents, examples and implications for C(κ, ω), including an extension of a result of P. Nyikos in nonmetric manifolds. Likewise for C(κ, κ) in Section 3. Section 4 develops more fully the relation to [α, β]-compact.