Characterizations of Strongly Paracompact Spaces

Theorem 3.1. A countably paracompact normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement.

Lemma 3.2 (see [2].)A space is strongly paracompact if and only if every increasing open cover of the space has a star-finite open refinement.

Lemma 3.3 (see [7].)Every countable open cover of a countably paracompact normal space has a star-finite open refinement.

Proof of Theorem 3.1. Necessity. By the definition of a strongly paracompact space, it is trivial to know that every increasing open cover of the space has a star-countable open refinement.

Sufficiency. Assume that X is a countably paracompact normal space and every increasing open cover of X has a star-countable open refinement. To prove that X is strongly paracompact, let 𝒪 be an increasing open cover of X and suppose that 𝒰 is a star-countable open refinement of 𝒪. With the help of Lemma 3.2, we prove that the cover 𝒪 has a star-finite open refinement.

Firstly, we present the family 𝒰 in the following way.

Claim. The family 𝒰 can be presented as 𝒰 = {ℬ_α : α ∈ Λ}, where each ℬ_α is a countable family and (⋃ℬ_α)⋂(⋃ℬ_β) = ∅ for α ≠ β.

Proof of claim. For all A, B ∈ 𝒰, we call the finite subfamily {C₁, C₂, …, C_n} a chain from A to B, if A = C₁, B = C_n, and C_i∩C_i+1 ≠ ∅ for 1 ≤ i < n. For every A ∈ 𝒰, denote

It is easy to know that ℬ(A) is countable, and, for any A₁, A₂ ∈ 𝒰, (⋃ℬ(A₁))⋂(⋃ℬ(A₂)) ≠ ∅ if and only if ℬ(A₁) = ℬ(A₂). We complete the proof of the claim.

For every α ∈ Λ, let Z_α = ∪ℬ_α. By the above claim, we know that the family {Z_α : α ∈ Λ} is an open and closed disjoint family of X. Since X is countably paracompact, the closed subspace Z_α of X is countably paracompact for every α ∈ Λ. Moreover, it follows from the above claim that the family ℬ_α is a countable open cover of Z_α. By Lemma 3.3, we find a star-finite open family 𝒲_α of the subspace Z_α refining ℬ_α. Since each Z_α is open in X and since X = ⋃_α∈ΛZ_α, it follows that the family ⋃_α∈Λ𝒲_α is an open cover of X. The family ⋃_α∈Λ𝒲_α is also star-finite since {Z_α : α ∈ Λ} is a disjoint family of X. On the other hand, it is easy to see that ⋃_α∈Λ𝒲_α is a refinement of 𝒪 since 𝒰 refines 𝒪.

By Lemma 3.2, the space X is strongly paracompact.

Remark 3.4. It is well known that space Γ constructed in [3] is not strongly paracompact, while every increasing open cover of the space has a star-countable refinement since it is linearly Lindelöf. It helps us to know that in Theorem 3.1 we cannot get the conclusion if we remove the countably paracompact property.

Corollary 3.5. Every perfectly normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement.

Proof. Necessity. It is trivial by the definition of strongly paracompact spaces.

Sufficiency. It is known that every perfectly normal space is countably paracompact and normal (see [5]). Then it follows from Theorem 3.1 that a perfectly normal space is strongly paracompact if every increasing open cover of the space has a star-countable open refinement.

Theorem 3.6. A space X is linearly D provided that every increasing open cover of X has a point-countable open refinement.

Proof. Assume that 𝒰 is an increasing open cover of X without a countable subcover, and 𝒱 is a point-countable open refinement of 𝒰.

In order to prove easily, well order X as {x_α : α < Γ} and let y₀ = x₀. Since 𝒱 is point countable, the family $$ is countable. For every $$, let U_V be the first set of 𝒰 such that U_V⊃V and let $$. The family 𝒰₀ cannot cover X since 𝒰₀ is a countable family and 𝒰 has no countable subcover according to our assumption above. We then take the first point of X which is not contained in ∪𝒰₀ and denote it by y₁. For every $$, let U_V be the first set of 𝒰 such that V ⊂ U_V. The family $$ is still not a cover of X. Consequently, we are able to take the first point of X which is not contained in ∪𝒰₁ and denote it by y₂. Thus, we continue to define the family $$, where each U_V is the first set of 𝒰 such that U_V⊃V. Define y_α and 𝒰_α successively in the same way. There must exist an ordinal Λ ≤ Γ such that the set A = {y_α : α < Λ} satisfies that the family 𝒰^′ = {U_V ∈ 𝒰 : V ∈ (𝒱) _A} covers X.

To prove that the set A is closed and discrete in X, it suffices to show that A is discrete since X is T₁. For every x ∈ X, if there exists some V^′ ∈ (𝒱) _x such that V^′∩A ≠ ∅, let β < Λ be the least such that y_β ∈ V^′∩A. Then $$ and y_α ∉ V^′ for every y_α with α < β. On the other hand, for every α > β, we know that y_α ∉ ∪𝒰_β and V^′ ∈ 𝒰_β, where $$. It follows that y_α ∉ V^′. Thus we have proved that such a neighborhood V^′ of x contains only one element of A. By the arbitrariness of x, we know that the set A is discrete.

To prove X is linearly D, it is enough to show that A is a 𝒰-big set. To show this, pick an arbitrary U ∈ 𝒰. Assume on the contrary that U^′ ⊂ U for every U^′ ∈ 𝒰^′. Then ∪𝒰^′ ⊂ U. It is contradicted with the fact that 𝒰 has no countable subcover. Therefore, there exists some U^′ ∈ 𝒰^′ such that U ⊂ U^′. Then, we have A⊄U. Thus we know that A is a 𝒰-big set.

We complete the proof of Theorem 3.6.

Corollary 3.7. A space of countable extent is linearly Lindelöf if and only if every increasing open cover of the space has a point-countable open refinement.

Lemma 3.8 (see [4].)Every monotonically normal linearly D-space is paracompact.

Theorem 3.9. A monotonically normal space X is strongly paracompact if and only if every increasing open cover of X has a star-countable open refinement.

Proof. Necessity. It is trivial by the definition of a strongly paracompact space.

Sufficiency. Assume that X is a monotonically normal space and every increasing open cover of X has a star-countable open refinement. It follows from Theorem 3.6 and Lemma 3.8 that X is paracompact. By Theorem 3.1, we know that X is strongly paracompact.