DOWKER SPACES REVISITED

LEWIS D. LUDWIG , PETER NYIKOS, AND JOHN E. PORTER

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

1. Introduction

A topological space X is called β-normal (α-normal) if for each pair of closed disjoint subsets A, B ⊂ X there are open sets U, V ⊂ X such that \overline{A ∩ U} = A, \overline{B ∩ V} = B and \overline{U} ∩\overline{V} = ∅ (U ∩V = ∅, respectively). This notion was introduced by Arhangel’skii and Ludwig in 1999 [1] and others have worked on the topic ([2], [6], [7], [9], [10], [11]) . In 1951, Dowker proved that a space X is countably paracompact and normal if and only if X × 𝕀 is normal [4]. A normal space X is called a Dowker space if X × 𝕀 is not normal, where 𝕀 is the unit interval with the usual topology. The main thrust of this article is to extend this work with regards α-normality and β-normality.

Section 2 is devoted to extending Dowker’s characterization of countably paracompact normal spaces to α-normal and β-normal spaces. The two main results of the section, Theorem 2.3 and Theorem 2.9, characterize when X × (ω + 1) is α-normal and when this product is β-normal. Corollaries 2.10 and 2.11 show what happens if in the forward supposition of Dowker’s theorem, normality is replaced with α-normality and β-normality (respectively). A new definition, α-countably paracompact, is introduced in this section and Corollary 2.7 shows that if X ×(ω+1) is β-normal, then X is β-normal and α-countably paracompact. The converse is an open question.

In Section 3, examples of Dowker spaces whose product with the unit interval are α-normal and β-normal (respectively) are given. Curiously, this section also exhibits Dowker spaces whose product with the unit interval are not α-normal and β-normal (respectively).

In Section 4 a collectionwise Hausdorff Moore space constructed by M. Wage is shown to be α-normal but not β-nornal. The article concludes with a list of open questions in Section 5. Throughout the paper, unless otherwise stated, a “space” is a T_{1}, regular, topological space. The ordinals ω, ω_{1} are used to denote the first two infinite cardinals. Readers may refer to Engelking [5] for undefined terms.

Date: November 1, 2006.

2000 Mathematics Subject Classification. 54D15.

Key words and phrases. α-normal, β-normal, products, Dowker, Moore spaces, α-countably paracompact.

2. Extending Dowker’s result to α-normality and β-normality

To start, we restate Dowker’s characterization of countably paracompact normal spaces as a fact for later reference purposes.

Fact 2.1. A topological space X is countably paracompact and normal if and only if X × 𝕀 is normal.

In light of Dowker’s characterization, it is natural to ask what would happen if one weakened the supposition that X × 𝕀 is normal to α-normal. We begin with a characterization of α-normal spaces. The proof is left to the reader.

Lemma 2.2. A topological space X is α-normal if and only if for every pair H and K of disjoint closed subsets of X there exists an open set U of X such that \overline{H ∩ U} = H and \overline{U} ∩ K is nowhere dense in K.

It should be noted, that in the standard proof of Fact 2.1, the reverse direction only uses the existence of a non-trivial convergent sequence in the space 𝕀 [5]. So we can actually say X × (ω + 1) is normal if and only if X is normal and countably paracompact. We now have the following.

Theorem 2.3. Let X be a T_{1} space. The product X × (ω + 1) is α-normal if and only if

(1) X is α-normal, and

(2) if {F_{n} : n ∈ ω} is a family of closed sets and F = ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} F_{k}), and E is

a closed subset of X disjoint from F, then there is a family {W_{n} : n ∈ ω} of

open sets such that W_{n} ∩F_{n} is dense in F_{n} and ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} W_{k}) is nowhere

dense in E.

Proof. Let {F_{n} : n ∈ ω} be a family of closed sets and F = ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} F_{k}), and

suppose E is a closed subset of X disjoint from F. Then A = (⋃{F_{n} × {n} : n ∈ ω}) ∪ (F × {ω}) and B = E × {ω} are disjoint closed subsets of X × (ω + 1). Since X × (ω + 1) is α-normal, there is an open subset W of X × (ω + 1) such that \overline{W ∩ A} = A and \overline{W} is nowhere dense in B.

For each n ∈ ω, define W_{n} = {x ∈ X : (x, n) ∈ W }. Clearly, W_{n} ∩ F_{n} is dense

in F_{n}. Since \overline{W} is nowhere dense in B, ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} W_{k}) is nowhere dense in E.

Conversely, let A and B be disjoint closed subsets of X × (ω + 1). Consider the sets

• A_{n} = {x ∈ X : (x, n) ∈ A},
• A_{ω} = {x ∈ X : (x, ω) ∈ A},
• B_{n} = {x ∈ X : (x, n) ∈ B}, and
• B_{ω} = {x ∈ X : (x, ω) ∈ B}.

Since X is α-normal and A_{ω} and B_{ω} are disjoint closed subsets of X, there are disjoint open subsets U_{A} and U_{B} of X such that cl_{X}(U_{A} ∩ A_{ω}) = A_{ω} and cl_{X}(U_{B} ∩ B_{ω}) = B_{ω}.

By (2) and the α-normality of X, there are open subsets U_{n} and V_{n} of X such that

(a) U_{n} ∩ A_{n} is dense in A_{n} for each n ∈ ω,

(b) ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} U_{k}) is nowhere dense in B_{ω},

(c) V_{n} is dense in B_{n} for each n ∈ ω,

(d) ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} V_{k}) is nowhere dense in A_{ω}, and

(e) U_{n} ∩ V_{n} = ∅ for each n ∈ ω.

Let H = A_{ω} \ ⋂_{n∈ω} cl_{X}(⋃_{k=n}^{∞} V_{k}) and K = B_{ω} \ ⋂_{n∈ω} cl_{X}(⋃_{k=n}^{∞} U_{k}). For each d ∈ H ∩ U_{A}, there is an open subset O_{d} and n_{d} ∈ ω such that

• O_{d} ⊂ U_{A}, and

• O_{d} ∩ ⋃_{k=n_{d}}^{∞} V_{k} = ∅.

Similarly, for each d ∈ K ∩ U_{B}, there is an open subset O_{d} and n_{d} ∈ ω such that • O_{d} ⊂ U_{B}, and

• O_{d} ∩ ⋃_{k=n_{d}}^{∞} U_{k} = ∅.

Let

U = ⋃{U_{n} × {n} : n ∈ ω} ∪ ⋃{O_{d} × [n_{d}, ω] : d ∈ H ∩ U_{A}}, and
V = ⋃{V_{n} × {n} : n ∈ ω} ∪ ⋃{O_{d} × [n_{d}, ω] : d ∈ K ∩ U_{B}}.

Note that U and V are disjoint subsets of X × (ω + 1), U is dense in A, and V is dense in B. Hence X × (ω + 1) is α-normal.

□

The β-normal case is similar to the α-normal case, albeit more complicated. For convenience, we break up the theorem into two parts.

Lemma 2.4. Let X be a T_{1} space. If X × (ω + 1) is β-normal, then:

(1) X is β-normal and

(2) if {F_{n} : n ∈ ω} is a family of closed sets and F = ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} F_{k}), and E is

a closed subset of X disjoint from F, then there is a family {W_{n} : n ∈ ω} of

open sets such that W_{n} ∩ F_{n} is dense in F_{n} and ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} W_{k}) is disjoint

from E.

Proof. Clearly X is β-normal. Let {F_{n} : n ∈ ω} be a family of closed sets with

F = ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} F_{k}), and E is a closed subset of X disjoint from F. Note that

A = ⋃_{n∈ω}(F_{n} ×{n})∪(F ×{ω}) and B = E ×{ω} are disjoint closed sets in X ×Y. Since X ×Y is β-normal, there are open U, V ⊂ X ×Y such that A∩U is dense in A, B∩V is dense in B, and \overline{U} ∩\overline{V} = ∅. The sets W_{n} = \{x ∈ X : (x, n) ∈ U\} are open in X and W_{n} ∩ F_{n} is dense in F_{n} for each n ∈ ω. Suppose C = ⋂_{n∈ω}(cl_{X}(⋃_{k=n}^{∞} W_{k}) is not disjoint from E. Let z ∈ C ∩ E. Then every neighborhood of z meets infinitely many W_{n} and since W_{n} × \{n\} ⊂ U, so (z, ω) ∈ \overline{U}. This is impossible since \overline{U} ∩ B = ∅. □

At this point, it should be noted that Dowker had a useful characterization of countably paracompact.

Fact 2.5. A topological space X is countably paracompact if and only if for every decreasing sequence ⟨F_{n} : n ∈ ω⟩ of closed subsets of X satisfying ⋂_{n∈ω} F_{n} = ∅ there exists a sequence ⟨W_{n} : n ∈ ω⟩ of open subsets of X such that F_{n} ⊂ W_{n} for n ∈ ω and ⋂_{n∈ω} \overline{W_{n}} = ∅.

This characterization prompted one of the authors to define the concept of αcountably metacompact and α-countably paracompact spaces.

Definition 2.6. A topological space is said to be α-countably paracompact (resp., α-countably metacompact) if for every decreasing sequence ⟨F_{n} : n ∈ ω⟩ of closed subsets of X satisfying ⋂_{n∈ω} F_{n} = ∅ there exists a sequence ⟨W_{n} : n ∈ ω⟩ of open subsets of X such that W_{n} ∩ F_{n} is dense in F_{n} for n ∈ ω and ⋂_{n∈ω} \overline{W_{n}} = ∅ (resp., ⋂_{n∈ω} W_{n} = ∅).

With this new definition and Lemma 2.4, we have the following corollary that exhibits what can be said of a space X if X × (ω + 1) is β-normal. In this direction, we can extend the result to the product of X and a compact metric space as all that is needed is a distinct convergent sequence and its limit point. The proofs are left to the reader.

Corollary 2.7. If X × (ω + 1) is β-normal, then X is β-normal and α-countably paracompact.

Corollary 2.8. If Y is an infinite compact metric space, and X × Y is β-normal, then X is β-normal and α-countably paracompact.

With Lemma 2.4 in hand, we are now ready for the main β-normal result of this section.

Theorem 2.9. Let X be a T_{1} space. The product X × (ω + 1) is β-normal if and only if the following three conditions are met:

(1) X is β-normal,

(2) condition (2) of Lemma 2.4 is satisfied, and

(3) for every decreasing sequence ⟨F_{n} : n ∈ ω⟩ of closed subsets of X satisfying ⋂_{n∈ω} F_{n} = ∅, there is a family {V_{n} : n ∈ ω} of open sets such that F_{n} ⊂ V_{n}

and ⋂_{n=0}^{∞} cl_{X}(V_{n}) is nowhere dense in the relative topology of F_{0}.

Proof. Let C be a closed subset of X × (ω + 1) and U an open set of X × (ω + 1) containing C. It suffices to find an open set G such that G ∩ C is dense in C and \overline{G} ⊂ U. Consider the following sets:

• C_{n} = {x ∈ X : (x, n) ∈ C}

• C_{ω} = {x ∈ X : (x, ω) ∈ C}
• U_{n} = {x ∈ X : (x, n) ∈ U}
• U_{ω} = {x ∈ X : (x, ω) ∈ U}

Note that each C_{n} and C_{ω} are closed subsets of X, and each U_{n} and U_{ω} are open

subsets of X. Also, F = ⋂_{n∈ω} cl_{X}( ⋃_{k=n}^{∞} C_{k}) is a closed subset of C_{ω}. By condition (2)

of Lemma 2.4 and β-normality of X, we can find open sets W_{n} such that

(a) ⋂_{n=0}^{∞} cl_{X}( ⋃_{k=n}^{∞} W_{k}) is a subset of U_{ω},

(b) W_{n} ∩ C_{n} is dense in C_{n}, and

(c) cl_{X}(W_{n}) ⊂ U_{n}.

Let G_{0} = ⋃_{n=0}^{∞}(W_{n} × \{n\}). Note that \overline{G_{0}} ∩ (X × \{ω\}) is a subset of X × (ω + 1),

while E = X \ U_{ω} is a subset of X. Hence \overline{G_{0}} ⊂ U.

By the β-normality ^{1} of X, we can find an open set V ⊂ X such that V ∩ C_{ω} is dense in C_{ω} and cl_{X}(V) ⊂ U_{ω}. Let E_{k} = X \ U_{k} and let F_{n} = cl_{X}(⋃_{k=n}^{∞} E_{k}) ∩ C_{ω}. Then ⟨F_{n} : n ∈ ω⟩ is a decreasing sequence of closed subsets of X such that ⋂_{i=1}^{∞} F_{i} = ∅ and by condition (3) of Theorem 2.9 we can find open sets V_{n} ⊃ F_{n} such that ⋂_{n∈ω} \overline{V_{n}} meets F_{0} in a nowhere dense set.

Now C_{ω} \V_{n} is a closed set disjoint from cl_{X}(⋃_{k=n}^{∞} E_{k}). So W_{n} = (V \\overline{V_{n}})×[n, ω] is an open set whose closure is disjoint from E_{k} for all k and so, by definition of V and E_{k}, the closure of W_{n} is a subset of U. Let G_{1} = ⋃_{n=0}^{∞} W_{n}. Then G_{1} ∩ C_{ω} = C_{ω} \ ⋂_{n=0}^{∞} \overline{V_{n}} is dense in C_{ω}, and the closure of G_{1} is easily seen to be a subset of U. Thus G = G_{0} ∪ G_{1} is the desired open set.

Conversely, let Y = X × (ω + 1) and suppose Y is It remains to verify condition (3) of Theorem 2.9.

Consider a decreasing sequence ⟨F_{n} : n ∈ ω⟩ of closed subsets of X satisfying ⋂_{i=1}^{∞} F_{i} = ∅. Let C = F_{0} × {ω} and E = ⋃_{n∈ω} F_{n} × {n}. Then C and E are disjoint closed subsets of Y. By β-normality of Y, there is an open subset W of Y whose intersection with C is dense in C, and whose closure is a subset of Y \ E. Let V_{n} = \{x ∈ X : (x, n) ∉ \overline{W} \}. Clearly, V_{n} is an open subset of X, and F_{n} ⊂ V_{n}. If ⋂_{n∈ω} \overline{V_{n}} ≠ ∅, let z be in the intersection. Every neighborhood of (z, ω) meets all but finitely many of the sets \overline{V_{n}} × \{n\}. Each of these sets is a subset of the closure of Y \ \overline{W} and so it misses W. Therefore, (z, ω) ∉ W, and so ⋂_{n∈ω} \overline{V_{n}} meets F_{0} in a nowhere dense subset of F_{0}. □

We now consider what happens if the supposition in Fact 2.1 is changed from X normal to X α-normal or β-normal. Notice that countably paracompact spaces satisfy condition (2) of Theorem 2.3 and conditions (2) and (3) of Theorem 2.9. Hence, we have the following corollaries.

Corollary 2.10. Let X be α-normal and countably paracompact. Then X ×(ω +1) is α-normal.

Corollary 2.11. Let X be β-normal and countably paracompact. Then X ×(ω +1) is β-normal.

1Here we use the equivalent definition of β-normality: A space X is β-normal if for each closed A ⊆ X and for every open U ⊆ X that contains A, there exists and open V ⊆ X such that \overline{V ∩ A} = A ⊆ \overline{V} ⊆ U.

3. Motivating Examples

After Dowker characterized countably paracompact normal spaces (Fact 2.1), he asked whether every normal space is countably paracompact or not. That is, does there exist a normal space X such that X × 𝕀 is not normal (i.e., a Dowker space). For several decades, the Dowker problem has fueled a great deal of research. In 1971, M.E. Rudin constructed a Dowker space [13]. In light of α-normality and β-normality, it is natural to ask whether the product of a Dowker space and the unit interval can be α-normal or β-normal.

Example 3.1. The product of a normal space and a compact metric space can be α-normal without being normal.

Proof. Consider a hereditarily separable Dowker space X and a compact metric space Y. Hereditarily separable Dowker spaces have been constructed under a variety of axioms independent of ZFC (see [12], [14], [15], and [16]). Since these spaces are hereditarily separable and Y is second countable, X × Y is hereditarily separable. A hereditarily separable regular space is α-normal [1]. □

We will see that the properties of the Dowker space dictate the outcome of the product. In Example 3.1, the product of a hereditarily separable Dowker space with a compact metric space resulted in an α-normal product space. If this condition is dropped, as the next example demonstrates, the product may fail to be α-normal.

Example 3.2. (ZFC) The product of a normal space and a compact metric space need not be α-normal.

Proof. Recall that a topological space X is called a P-space if the intersection of countably many open sets is open. If X is a Dowker P -space and is extremally disconnected, that is if the closure of an open set is open, then X fails condition (2) of Theorem 2.3. Dow and van Mill [3] have constructed such a space in ZFC. □

Remark 3.3. Note that in extremally disconnected spaces, α-countably paracompactness is equivalent to countable paracompactness. Thus, the normal space in Example 3.2 is not α-countably paracompact.

Although β-normality seems a much stronger condition than α-normality, it is not enough to determine the Dowker situation as the next example illustrate.

Example 3.4. (V=L) A normal space whose product with a compact metric space is β-normal but not normal.

Proof. P. Nyikos [12] constructed a scattered hereditarily strongly collectionwise (scwH) Hausdorff Dowker space X under the axiom V=L. Recently, it was shown that X × (ω + 1) is scattered and hereditarily scwH, and therefore X × (ω + 1) is (hereditarily) β-normal by Nyikos and Porter’s Theorem 2.8 [11]. □

4. Moore space results

In light of Theorem 2.9, one may consider how close α-normal and β-normal are in the presence of condition (2) of Theorem 2.3 and conditions (2) and (3) of Theorem 2.9. The next example gives some insight on this.

Example 4.1. A first countable Tychonov space that is α-normal, collectionwise Hausdorff, and α-countably paracompact but not β-normal.

Proof. In [17], Wage produced an example of a collectionwise Hausdorff first countable Tychonoff space that is not normal. We state the following lemma used by Wage.

Lemma 4.2. There exist subsets of the real line A and B such that B ⊂ A and every countable subset of B is contained in a G_{δ} that does not meet A − B, yet every G_{δ} containing B does meet A − B.

To create Wage’s example, topologize A by letting the points of B have the usual neighborhoods and each point of A − B be isolated. Let X = A × (ω + 1) − B × {ω}. Wage showed this space to be first countable, Tychonoff, non-normal, pseudonormal, and collectionwise Hausdorff. A similar argument to the one Wage used to show that X is collectionwise Hausdorff will be used to show that X is α-normal.

Since the points of (A − B) × ω are isolated and B × ω is hereditary separable, it suffices to show that every countable subset of B × ω is contained in an open set whose closure misses (A − B) × {ω}. Let C ⊂ B × ω be countable. Since every countable subset of B is contained in a G_{δ} that misses A − B, there exist open sets U_{n} ⊂ A with U_{n+1} ⊂ U_{n} such that {x ∈ B : (∃n ∈ ω)(x, n) ∈ C} ⊂ ⋂{U_{n} : n ∈ ω} and ⋂{U_{n} : n ∈ ω} ∩ (A − B) = ∅. Note that C ⊂ ⋃{U_{n} × {n} : n ∈ ω}, and the closure of ⋃{U_{n} × {n} : n ∈ ω} misses (A − B) × {ω}. That is, X is α-normal.

To show that X is α-countably paracompact, let {F_{n} : n ∈ ω} be a sequence of decreasing closed sets such that ⋂ F_{n} = ∅. For each n ∈ ω let

(i) G_{n} = F_{n} ∩ (B × ω), (ii) H_{n} = F_{n} ∩ (A − B) × {ω}, and (iii) I_{n} = F_{n} ∩ (A − B) × ω

Note that I_{n} is open in X. For each (x, ω) ∈ H_{n}, let U_{(x,ω)} = {(x, k) : k ≥ n}. Note that \overline{⋃\{U_{(x,ω)} : (x, ω) ∈ H_{n}\}} ∩ (A − B) × \{ω\} = H_{n}. Since A × ω is paracompact open subset of X and {G_{n} : n ∈ ω} is a nested sequence of closed sets, we can find open sets V_{n} such that G_{n} ⊂ V_{n} and ⋂ \overline{V_{n}} ∩ (A × ω) = ∅. Since G_{n} is closed, we can find an open set O_{n} such that \overline{G_{n} ∩ O_{n}} = G_{n} and \overline{O_{n}} ∩ (A − B) × \{ω\} = ∅ by the above arguments. Let U_{n} = V_{n} ∩ O_{n}, and let

W_{n} = I_{n} ∪ (⋃{U_{(x,ω)} : (x, ω) ∈ H_{n}}) ∪ U_{n}.

Note that ⋂ \overline{W_{n}} = ∅, and X is α-countably paracompact.

To show that X is not β-normal, we show that the closed sets B×ω and (A−B)× {ω} cannot be β-separated. Suppose U and V are open sets such that (B ×ω)∩U is dense in B ×ω and (A−B)×{ω}∩V is dense in (A−B)×{ω}. Since (A−B)×{ω} is discrete, for every x ∈ A − B there is an n_{x} ∈ ω such that {(x, n) : n ≥ n_{x}} ⊂ V. We claim there exists x′ ∈ B and a sequence {x_{k}} in A − B and an m ∈ ω such that x_{k} → x′ and n_{x_{k}} = m. Since \overline{U} ⊃ B × ω, \overline{U} must contain \{(x′, n) : n ∈ ω\}. This shows that \overline{U} ∩ \overline{V} ≠ ∅.

To prove the claim, let E_{m} = {x ∈ A − B : n_{x} = m}. If the claim were not true, then for every x ∈ B there is a neighborhood O_{x} of x such that O_{x} ∩ E_{m} = ∅. Let O_{m} = ⋃_{x∈X} O_{x}. Note that O_{m} ∩E_{m} = ∅, and ⋂_{m∈ω} O_{m} is a G_{δ} set which contains B but misses A − B, a contradiction. This completes the proof. □

Wage used this space to construct a collectionwise Hausdorff non-normal Moore space. This gives the following interesting result.

Example 4.3. There exists a collectionwise Hausdorff, non-β-normal, α-normal Moore space, X′.

Proof. Let X′ be the set of all non-isolated points of X. The space

Y = X′ × {ω} ∪ (X − X′) × ω

as a subspace of X × (ω + 1) is a Moore space which is α-normal but not β-normal by the previous arguments. □

5. Questions

The authors close the paper by listing some open questions that the authors were unable to answer.

Question 5.1. Is there a Dowker space whose product with a compact metric space is β-normal in ZFC?

Question 5.2. Is there a Dowker space whose product with a compact metric space is α-normal, but not β-normal?

Question 5.3. If X is β-normal and α-countably paracompact, is X × (ω + 1) β-normal? α-normal?

Question 5.4. If X is α-normal and α-countably paracompact, is X × (ω + 1) α-normal?

Question 5.5. If X × (ω + 1) is α-normal, is X α-countably metacompact? αcountably paracompact?

Question 5.6. Are β-normal α-countably metacompact spaces α-countably paracompact?

Question 5.7. Is there a β-normal non-normal Moore space?

It follows from condition (3) of Theorem 2.9 that the product of a Dowker P - space with ω + 1 is not β-normal. Recall that Rudin’s Dowker [13] space is a P -space.

Question 5.8. Is Rudin’s Dowker space α-countably paracompact?

Question 5.9. Is the product of Rudin’s Dowker space with ω + 1 α-normal?

References

[1] A. Arhangel’skii, L. Ludwig, On α-normal and β-normal spaces, Comment. Math. Univ. Carolin., 42 (2001), no. 3, 507–519.

[2] D. Burke, L. Ludwig Hereditarily α-normal spaces and infinite products., Topology Proc. 25 (2000), Summer, 291–299.

[3] A. Dow, J. van Mill, An extremally disconnected Dowker space, Proc. Amer. Math. Soc., 86 (1982), no. 4, 669–672.

[4] C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951), 219-224.

[5] R. Engelking, General topology, second ed., Heldermann Verlag, Berlin, 1989.

[6] Y. Hiraya, N. Kemoto, The hereditarily collectionwise Hausdorff property in products of ω_{1}, Topology Proc. 29 (2005), no. 1, 167–173.

[7] L. Kalantan, Results about κ-normality, Topology Appl., 125 (2002), no. 1, 47-62.

[8] L. Ludwig, Two generalizations of normality: α-normality and β-normality, Ph.D. dissertation, Ohio University, 2001.

[9] L. Ludwig and P. Szeptycki, A consistent example of a β-normal, non-normal space, Topology Proc., 25, (2000), Spring, 251–254.

[10] E. Murtinová, A β-normal Tychonoff space which is not normal, Comment. Math. Univ. Carolin., 43 (2002), no. 1, 159-164.

[11] P. Nyikos, J. E. Porter Hereditarily strongly cwH and separation axioms, to appear in Topology Appl.

[12] P. Nyikos, Hereditarily normal, locally compact Dowker spaces, Topology Proceedings 24 (1999) 261–276.

[13] M. E. Rudin, A normal space X for which X × I is not normal, Fund. Math., 73, 1971, no. 2, 179–186.

[14] M. E. Rudin, A separable Dowker space, Symposia Mathematica, Vol. XVI (Convegno sulla Topologia Insiemistica e Generale, INDAM, Roma, Marzo, 1973), pp. 125–132. Academic Press, London, 1974.

[15] M. E. Rudin, Dowker Spaces, Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, (1984), 761-780.

[16] M. E. Rudin, Countable point separating open covers for manifolds, Houston J. Math. 15 (1989), no. 2, 255–266.

[17] M. Wage, A Collectionwise Hausdorff, Non-normal Moore Space, Canad. J. Math. 3 (1976) 632–634.

Department of Mathematics and Computer Science, Denison University, Granville, OH 43023, LUDWIGL@DENISON.EDU

Department of Mathematics, University of South Carolina, Columbia, SC 29208, NYIKOS@MATH.SC.EDU

Department of Mathematics and Statistics, Faculty Hall 6C, Murray State University, Murray, Ky 42071, TED.PORTER@MURRAYSTATE.EDU