ON 3-TOPOLOGICAL VERSION OF Θ-REGULARITY

We modify the concept of θ-regularity for spaces with 2 and 3 topologies. The new, more general property is fully preserved by sums and products. Using some bitopological reductions of this property, Michael’s theorem for several variants of bitopological paracompactness is proved.

1. Preliminaries.The term space (X,τ,σ ,ρ) is referred as a set X with three, generally nonidentical topologies τ, σ , and ρ.We say that x ∈ X is a (σ , ρ)-θ-cluster point of a filter base Φ in X if for every V ∈ σ such that x ∈ V and every F ∈ Φ the intersection F ∩ cl ρ V is nonempty.If, for every V ∈ σ with x ∈ V , there is some F ∈ Φ with F ⊆ cl ρ V , we say that Φ(σ , ρ)-θ-converges to x.Then x is called a (σ , ρ)-θ-limit of Φ.If Φ converges or has a cluster point with respect to the topology τ, we say that Φ τ-converges or has a τ-cluster point.
A family is called σ -locally finite if it consists of countably many locally finite subfamilies.(This notion has nothing common with the topology also denoted by σ .)For a family Φ ⊆ 2 X , we denote by Φ F the family of all finite unions of members of Φ.A family Φ is called directed if Φ F is a refinement of Φ.
We say that the space (X,τ,σ ,ρ) is (τ − σ ) (semi-) paracompact with respect to ρ if every τ-open cover of X has a σ -open refinement which is (σ -) locally finite with respect to the topology ρ.
Recall that the topological space (X, τ) is called (countably) θ-regular [2] if every (countable) filter base in (X, τ) with a θ-cluster point has a cluster point.

Main results
Theorem 2.1.Let τ, σ , ρ be topologies on X.The following statements are equivalent: (i) For every (countable) τ-open cover Ω of X and each x ∈ X there is a σ -open neighborhood U of x such that cl ρ U can be covered by a finite subfamily of Ω.
(iv) For every (countable) filter base Φ in X with no τ-cluster point and every x ∈ X there are U ∈ σ , V ∈ ρ, and F ∈ Φ such that x ∈ U, F ⊆ V , and U ∩ V = ∅.
Similarly as for θ-regularity, there are numbers of simple examples of (τ,σ ,ρ)θ-regular spaces, including various modifications of regularity, compactness, local compactness, or paracompactness and we leave them to the reader.Note, for example, that a space (τ − σ ) paracompact with respect to ρ is (τ,ρ,σ )-θ-regular.Remark 2.4.One can easily check that (τ,σ ,ρ)-θ-regularity is preserved by τclosed subspaces if we consider the corresponding induced topologies on the subspace.On the other hand, as it is shown in [3], even F σ -subspace of a compact (non-Hausdorff) space need not be countably θ-regular.
However, for the following bitopological modifications of well-known Michael's theorem [5], only the β-and δ-versions of pairwise (countable) θ-regularity will be useful.In the proof of the next theorem, we slightly modify the technique used in [3].

Proof.
Let Ω be a σ 1 -open cover of X.Since X is (σ 1 − σ 2 ) semiparacompact with respect to σ 3 , it follows that Ω has a σ 2 -open refinement, say Ω = ∞ i=1 Ω i , where every Ω i is a locally finite with respect to σ 3 family refining Ω. Let semiparacompact with respect to σ 2 , Φ has a σ 3 -open refinement, say Φ = ∞ i=1 Φ i , consisting of families Φ i which are locally finite with respect to σ 2 .For every n ∈ N, let The family {V n } n∈N is a σ 3 -open increasing cover of X.Because the family n i=1 Φ i is locally finite with respect to σ 2 , we have cl σ 2 V n ⊆ U n−1 .Finally, for every n ∈ N and U ∈ Ω n , let (2. 2) It can be easily seen that the family open cover of X which is a refinement of Ω locally finite with respect to σ 3 .Indeed, for every x ∈ X let k ∈ N be the least index such that x ∈ U for some open cover which, obviously, refines Ω.To see that Γ is locally finite with respect to σ 3 , let x ∈ X and let m ∈ N be any index such that x ∈ V m .Because {V n } n∈N is an increasing family, we have But the family m i=1 Ω i is locally finite with respect to σ 3 .Let S be a σ 3 -neighborhood of x, intersecting at most finitely many elements of m i=1 Ω i .Since for every i = 1, 2,...,m, U ∈ Ω i , we have W i (U ) ⊆ U , the set S ∩ V m is a σ 3 -neighborhood of x, meeting only finitely many sets of the cover Γ .Hence Γ is locally finite with respect to σ 3 and therefore X is (σ 1 − σ 2 ) paracompact with respect to σ 3 .
In order to obtain a theorem for a bitopological space (X, τ 1 ,τ 2 ) from Theorem 2.10 it can be easily seen that there are only three meaningful possibilities for identifying the topologies σ 1 , σ 2 , σ 3 , σ 4 .
Proof.It is sufficient to use the previous corollary twice.
Note that Raghavan and Reilly stated [7, Theorem 3.9] from which it would follow that a pairwise regular δ-pairwise semiparacompact space is δ-pairwise paracompact.Unfortunately, (iv) ⇒(i) in the proof of this theorem is not correct.The authors used [1, Theorem 1.5, page 162] in the proof.However, the assumptions of the theorem are not completely satisfied.They tried to expand a locally finite cover ᐂ to the open one using a closed cover such that every its element meets only finitely many members of ᐂ.However, in general the used closed cover is not locally finite or at least closure preserving.That is not sufficient for the expansion, as the following example shows.On the other hand, the previous example does not refute Raghavan-Reilly's theorem, which still remains open as a question.With a different modification of the concept of pairwise regularity the theorem is correct.
Remark 2.15.Note that the space X constructed in Example 2.13 is T 3.5 but not normal-the sets A, X C are closed, pairwise disjoint but they have no disjoint neighborhoods.
Corollary 2.16.Let X be a bitopological space.Then X is RR-pairwise paracompact if and only if X is β-pairwise countably θ-regular and RR-pairwise semiparacompact.
Finally, remark that modifying properly the concept of -space for bitopological spaces, combining Theorem 2.6 and the corollaries of Theorem 2.10 similar results as in [4] (see [6, Nagami's theorem]) for the countable product of paracompact -spaces without necessity of Hausdorff-type separation are also possible.

Example 2 .
13. Let C = N × −1, 1), B = N × (0, 1), and A = N × −1, 0 .We consider the Euclidean topology on C induced from the real plane and letX = C ∪ {y | y is a nonconvergent ultra-closed filter in C, B ∈ y}.Let S(U) = U ∪ {y | y ∈ X C, U ∈ y} for any U ⊆ C open in C.Of course, X is a subspace of the Wallman compactification ωC and the sets S(U) constitute a topology base for X.Since C is normal, ωC is Hausdorff and hence X is a T 3.5 space.Denote A n = {n} × −1, 0 .The familyΩ = S(B), A 1 ,A 2 ,A 3 ,...is a locally finite cover of X, which has no open locally finite extension.Indeed, suppose that there are some open U n with A n ⊆ U n for n ∈ N. Then every U n must meet B n = {n} × (0, 1).Choose x n ∈ U n ∩ B n for each n ∈ N. Let F n = {x n ,x n+1 ,...}.Since the sequence x 1 ,x 2 ,... has no cluster point in C, the collection Φ = {F n | n = 1, 2,...} is a closed filter base in C with no cluster point in C. It follows that there is a non-convergent ultra-closed filter, say y ∈ ωC, finer than Φ.But F 1 ⊆ B and since F 1 ∈ Φ ⊆ y, B ∈ y.Hence y ∈ X.Let W be any open neighborhood of y in X.There is some V open in C with y ∈ S(V ) ⊆ W . Then V ∈ y and hence V ∩ F n ≠ ∅ for every n ∈ N. Thus for any fixed m ∈ N there exists n ≥ m such that x n ∈ V ⊆ S(V ) ⊆ W and therefore W intersects infinitely many elements of {U n | n = 1, 2,...}.Hence Ω cannot be expanded to an open locally finite cover.