Mathematical Problems in Engineering

Vol. 23, No. 6 (2000) 393–398 S0161171200001678

© Hindawi Publishing Corp.

ON 3-TOPOLOGICAL VERSION OF Θ-REGULARITY

MARTIN M. KOVÁR (Received 14 July 1998)

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

Keywords and phrases. Spaces with 2 and 3 topologies,θ-regularity, paracompactness.

2000 Mathematics Subject Classification. Primary 54E55, 54A10; Secondary 54D20, 54A20.

1. Preliminaries. The termspace(X,τ,σ ,ρ)is referred as a setXwith three, gen- erally nonidentical topologiesτ,σ, andρ. We say thatx∈Xis a(σ ,ρ)-θ-cluster point of a filter baseΦinXif for everyV∈σ such thatx∈Vand everyF∈Φthe intersec- tionF∩clρV is nonempty. If, for everyV∈σ withx∈V, there is someF∈Φwith F ⊆clρV, we say thatΦ(σ ,ρ)-θ-convergestox. Thenx is called a(σ ,ρ)-θ-limit of Φ. IfΦconverges or has a cluster point with respect to the topologyτ, we say thatΦ τ-convergesor has aτ-cluster point.

A family is calledσ-locally finiteif it consists of countably many locally finite sub- families. (This notion has nothing common with the topology also denoted byσ.) For a familyΦ⊆2^X, we denote byΦ^F the family of all finite unions of members ofΦ. A familyΦis calleddirectedifΦ^F is a refinement ofΦ.

We say that the space(X,τ,σ ,ρ)is(τ−σ )(semi-) paracompact with respect toρ if everyτ-open cover ofXhas a σ-open refinement which is (σ-) locally finite with respect to the topologyρ.

The bitopological space(X,τ,σ )is called RR-pairwise (semi-) paracompact if the space is(τ−τ)(semi-) paracompact with respect toσand(σ−σ )(semi-) paracompact with respect toτ. We say that(X,τ,σ )isFHP-pairwise (semi-) paracompactif the space is(τ−σ )(semi-) paracompact with respect toσand(σ−τ)(semi-) paracompact with respect toτ. Finally,(X,τ,σ )is said to beδ-pairwise (semi-) paracompactif the space is(τ−(τ∨σ ))(semi-) paracompact with respect toτ∨σ and(σ−(τ∨σ ))(semi-) paracompact with respect toτ∨σ (see [7]).

Recall that the topological space(X,τ)is called (countably)θ-regular [2] if every (countable) filter base in(X,τ)with aθ-cluster point has a cluster point.

2. Main results

Theorem2.1. Letτ,σ,ρbe topologies onX. The following statements are equiva- lent:

(i) For every (countable)τ-open coverΩ ofX and eachx∈X there is aσ-open

neighborhoodUofxsuch thatclρUcan be covered by a finite subfamily ofΩ.

(ii) Every (countable)τ-closed filter base Φ with a (σ ,ρ)-θ-cluster point has a τ- cluster point.

(iii) Every (countable) filter baseΦwith a(σ ,ρ)-θ-cluster point has aτ-cluster point.

(iv) For every (countable) filter baseΦinXwith noτ-cluster point and everyx∈X there areU∈σ,V∈ρ, andF∈Φsuch thatx∈U,F⊆V, andU∩V= ∅.

Proof. Suppose (i). LetΦbe a (countable) filter base inXwith noτ-cluster point.

ThenΩ= {X clτF |F ∈Φ}is a (countable)τ-open directed cover ofX. Letx∈X.

By (i) there is U∈σ with x∈U and clρU⊆X clτF for some F ∈Φ. DenoteV= X clρU. Thenx∈U,F⊆V∈ρandU∩V= ∅. It follows (iv).

The implications (iv)⇒(iii)⇒(ii) are clear. Suppose (ii). Take any (countable)τ-open coverΩofX. ThenΦ= {X V|V∈Ω^F}is aτ-closed filter base inXwith noτ-cluster point. Letx∈X. It follows from (ii) thatxis not a(σ ,ρ)-θ-cluster point ofΦ, so there are someU∈σ andV∈Ω^F such thatx∈Uand(X V)∩clρU= ∅. Then clρU⊆V, which implies (i).

Definition2.2. Letτ,σ,ρbe topologies onX. Then(X,τ,σ ,ρ)is said to be(count- ably)(τ,σ ,ρ)-θ-regular, ifXsatisfies any of the conditions (i)–(iv) of Theorem 2.1.

Note that forτ=σ=ρwe obtain the notion of (countably)θ-regular space. Omitting the condition of countability, we get further criteria of(τ,σ ,ρ)-θ-regularity.

Theorem2.3. Letτ,σ,ρbe topologies onX. The following statements are equiva- lent:

(i) Xis(τ,σ ,ρ)-θ-regular.

(ii) Every(σ ,ρ)-θ-convergent filter baseΦhas aτ-cluster point.

(iii) Every(σ ,ρ)-θ-convergent ultrafilter inXisτ-convergent.

Proof. The implications (i)⇒(ii)⇒(iii) are clear. Conversely, suppose (iii) and take a filter baseΦinXwith a(σ ,ρ)-θ-cluster pointx∈X. Letζbe aσ-open local base ofx. Then the familyΦ= {F∩clρV|F ∈Φ,V∈ζ}is a filter base finer thanΦand (σ ,ρ)-θ-converging to x. Denote byΓ an ultrafilter finer than Φ. ThenΦ⊆Γ and henceΓ also(σ ,ρ)-θ-converges tox. By (iii),Γ isτ-convergent to somey∈X. Since Γ is finer thanΦ,yis aτ-cluster point ofΦ.

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 toρis(τ,ρ,σ )-θ-regular.

Remark2.4. One can easily check that(τ,σ ,ρ)-θ-regularity is preserved by τ- closed subspaces if we consider the corresponding induced topologies on the sub- space. On the other hand, as it is shown in [3], evenFσ-subspace of a compact (non- Hausdorff) space need not be countablyθ-regular.

For a family {(Xι,τι,σι,ρι) | ι ∈ I} denote by τ,σ ,ρ the corresponding sum (product) topologies onX=

ι∈IXι(X=Πι∈IXι). It is an easy exercise to prove that the topological sumXof(τι,σι,ρι)-θ-regular spacesXι, whereι∈I, is(τ,σ ,ρ)-θ-regular.

Theorem2.5. LetX=

ι∈IXιbe the sum space for the family{(Xι,τι,σι,ρι)|ι∈I}

with the corresponding sum topologiesτ,σ ,ρ. Suppose that everyXιis(τι,σι,ρι)-θ- regular. ThenXis(τ,σ ,ρ)-θ-regular.

Theorem2.6. LetX=Πι∈IXιbe the product space for the family

(Xι,τι,σι,ρι)| ι∈I

with the corresponding product topologies τ, σ, ρ. Suppose that everyXι is (τι,σι,ρι)-θ-regular. ThenXis(τ,σ ,ρ)-θ-regular.

Proof. Let Γ be an ultrafilter inX with (σ ,ρ)-θ-limit x =(xι)ι∈I ∈X. Let πι: X→Xιbe the canonical projection. Thenπι(Γ)is an ultrafilter onXιwhich(σι,ρι)- θ-converges toxι. ButXιis(τι,σι,ρι)-θ-regular. Hence,πι(Γ) τι-converges to some yι∈Xι, which implies thatΓ τ-converges toy=(yι)ι∈I. It follows thatXis(τ,σ ,ρ)- θ-regular.

The productivity ofθ-regularity proved in [4] by a different technique now follows as a corollary.

Definition2.7. A bitopological space(X,τ,σ )is said to beα-pairwise (countably) θ-regular if X is (countably) (τ,τ,σ )-θ-regular and (countably) (σ ,σ ,τ)-θ-regular, β-pairwise (countably)θ-regular if X is (countably) (τ,σ ,τ)-θ-regular and (count- ably)(σ ,τ,σ )-θ-regular,γ-pairwise (countably)θ-regularifXis (countably)(τ,σ ,σ )- θ-regular and (countably) (σ ,τ,τ)-θ-regular and finally, δ-pairwise (countably) θ- regularifXis (countably)(τ∨σ ,σ ,τ∨σ )-θ-regular and (countably)(τ∨σ ,τ,τ∨σ )- θ-regular.

Remark2.8. Using the characterization (i) in Theorem 2.1 and refining the open covers of the space several times, one can easily check thatβ- andγ-versions of pair- wiseθ-regularity are equivalent and imply theα-version, but not vice versa. Since every pairwise regular space obviously isα-pairwiseθ-regular, the real line topologized by the intervals(−∞,p),p∈Rforτand(q,∞),q∈Rforσis a proper counterexample.

Remark2.9. Observe that RR-pairwise paracompact and FHP-pairwise paracom- pact spaces are β-pairwiseθ-regular and it can be easily seen that aβ-pairwiseθ- regular space has both topologiesθ-regular.

However, for the following bitopological modifications of well-known Michael’s the- orem [5], only theβ- andδ-versions of pairwise (countable)θ-regularity will be useful.

In the proof of the next theorem, we slightly modify the technique used in [3].

Theorem2.10. Letσ1, σ2, σ3, σ4, be topologies on X. LetX be (σ1−σ2)semi- paracompact with respect toσ3, (σ4−σ3)semiparacompact with respect to σ2 and countably(σ2,σ4,σ2)-θ-regular. ThenXis(σ1−σ2)paracompact with respect toσ3.

Proof. LetΩbe aσ1-open cover ofX. SinceXis(σ1−σ2)semiparacompact with respect toσ3, it follows that Ωhas a σ2-open refinement, say Ω=_∞

i=1Ωi, where everyΩiis a locally finite with respect toσ3family refiningΩ.

LetUn=

{U|U∈Ωi, i≤n}for everyn∈N. The family{Un}n∈N is a countable σ2-open increasing cover ofX and sinceXis countably(σ2,σ4,σ2)-θ-regular, there exists aσ4-open coverΦofXwhoseσ2-closures refine{Un}n∈N. BecauseXis(σ4−σ3)

semiparacompact with respect toσ2, Φhas aσ3-open refinement, sayΦ=_∞

i=1Φi, consisting of familiesΦiwhich are locally finite with respect toσ2. For everyn∈N, let

Vn=

B|B∈Φi,clσ2B⊆Uj, i+j≤n

. (2.1)

The family{Vn}n∈Nis aσ3-open increasing cover ofX. Because the family_n

i=1Φi

is locally finite with respect toσ2, we have clσ2Vn⊆Un−1. Finally, for everyn∈Nand U∈Ωn, let

Wn(U)=U clσ₂Vn. (2.2)

It can be easily seen that the familyΓ=

Wn(U)|n∈N, U∈Ωn

is aσ2-open cover ofXwhich is a refinement ofΩlocally finite with respect toσ3. Indeed, for everyx∈X letk∈Nbe the least index such thatx∈U for someU∈Ωk. Since clσ2Vk⊆Uk−1, it follows thatx∈Wk(U). HenceΓ is aσ2-open cover which, obviously, refinesΩ. To see thatΓ is locally finite with respect toσ3, letx∈X and letm∈Nbe any index such thatx∈Vm. Because{Vn}n∈Nis an increasing family, we haveVm∩Wn(U)= ∅ for everyn≥m,U∈Ωn.

But the family_m

i=1Ωiis locally finite with respect toσ3. LetSbe aσ3-neighborhood of x, intersecting at most finitely many elements of _m

i=1Ωi. Since for every i= 1,2,...,m, U∈Ωi, we haveWi(U)⊆U, the setS∩Vm is aσ3-neighborhood ofx, meeting only finitely many sets of the coverΓ. HenceΓ is locally finite with respect to σ3and thereforeXis(σ1−σ2)paracompact with respect toσ3.

In order to obtain a theorem for a bitopological space(X,τ1,τ2)from Theorem 2.10 it can be easily seen that there are only three meaningful possibilities for identifying the topologiesσ1,σ2,σ3,σ4.

Case(i). τ1=σ1=σ4andτ2=σ2=σ3.

Corollary2.11. LetXbe countably(τ2,τ1,τ2)-θ-regular and(τ1−τ2)semipara- compact with respect toτ2. ThenXis(τ1−τ2)paracompact with respect toτ2.

Corollary2.12. LetXbe a bitopological space. ThenXis FHP-pairwise paracom- pact ifand only ifXisβ-pairwise countablyθ-regular and FHP-pairwise semiparacom- pact.

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 regularδ-pairwise semiparacompact space isδ-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 coverᐂto the open one using a closed cover such that every its element meets only finitely many members ofᐂ. 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.

Example2.13. Let C=N× −1,1), B=N×(0,1), and A=N× −1,0. We con- sider the Euclidean topology onC induced from the real plane and letX=C∪ {y|

yis a nonconvergent ultra-closed filter inC, B∈y}. LetS(U)=U∪ {y|y∈X C, U∈y}for anyU⊆Copen inC. Of course,Xis a subspace of the Wallman compact- ificationωCand the setsS(U)constitute a topology base forX. SinceC is normal, ωCis Hausdorff and henceXis aT3.5space. DenoteAn= {n} × −1,0. The family Ω=

S(B),A1,A2,A3,...

is a locally finite cover ofX, which has no open locally finite extension.

Indeed, suppose that there are some openUn withAn⊆Unforn∈N. Then every Un must meet Bn = {n} ×(0,1). Choose xn ∈Un∩Bn for each n∈ N. Let Fn = {xn,xn+1,...}. Since the sequencex1,x2,... has no cluster point inC, the collection Φ= {Fn|n=1,2,...}is a closed filter base inCwith no cluster point inC. It follows that there is a non-convergent ultra-closed filter, sayy∈ωC, finer thanΦ. ButF1⊆B and sinceF1∈Φ⊆y,B∈y. Hencey∈X. LetWbe any open neighborhood ofyinX.

There is someVopen inCwithy∈S(V )⊆W. ThenV∈yand henceV∩Fn≠∅for everyn∈N. Thus for any fixedm∈Nthere existsn≥msuch thatxn∈V⊆S(V )⊆W and thereforeW intersects infinitely many elements of{Un|n=1,2,...}. HenceΩ cannot be expanded to an open locally finite cover.

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.

Corollary2.14. LetX be δ-pairwise countably θ-regular. ThenX is δ-pairwise paracompact ifand only ifXisδ-pairwise semiparacompact.

Proof. SinceX is countably (τ1∨τ2,τ1,τ1∨τ2)-θ-regular and (τ1−(τ1∨τ2)) semiparacompact with respect toτ1∨τ2, it follows thatXis(τ1−(τ1∨τ2))paracom- pact with respect toτ1∨τ2by Corollary 2.11. ButXis also countably(τ1∨τ2,τ2,τ1∨ τ2)-θ-regular and(τ2−(τ1∨τ2))semiparacompact with respect toτ1∨τ2which im- plies, also by Corollary 2.11, thatX is(τ2−(τ1∨τ2))paracompact with respect to τ1∨τ2. HenceXisδ-pairwise paracompact in topologiesτ1,τ2.

Remark2.15. Note that the spaceXconstructed in Example 2.13 isT3.5but not normal—the setsA,X Care closed, pairwise disjoint but they have no disjoint neigh- borhoods.

Case(ii). τ1=σ1=σ2andτ2=σ3=σ4.

Corollary2.16. LetXbe a bitopological space. ThenXis RR-pairwise paracom- pact ifand only ifXisβ-pairwise countablyθ-regular and RR-pairwise semiparacom- pact.

Case(iii). τ1=σ1=σ3andτ2=σ2=σ4.

Corollary2.17. Letτ1,τ2 be countablyθ-regular topologies of X. Suppose that X is (τ1−τ2) semiparacompact with respect toτ1, and(τ2−τ1)semiparacompact with respect toτ2. ThenXis(τ1−τ2)paracompact with respect toτ1and(τ2−τ1) paracompact with respect toτ2.

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.

Acknowledgement. This research is supported by grant GA ˘CR 201/97/0216.

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Kovár: Department of Mathematics, Faculty of Electrical Engineering and Com- puter Science, Technical University of Brno, Technická8, 616 69Brno, Czech Republic

E-mail address:[email protected]

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