A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality
aa r X i v : . [ m a t h . GN ] J u l A COMMON EXTENSION OF ARHANGEL’SKII’STHEOREM AND THE HAJNAL-JUH ´ASZ INEQUALITY
ANGELO BELLA AND SANTI SPADARO
Abstract.
We present a bound for the weak Lindel¨of numberof the G δ -modification of a Hausdorff space which implies variousknown cardinal inequalities, including the following two fundamen-tal results in the theory of cardinal invariants in topology: | X | ≤ L ( X ) χ ( X ) (Arhangel’ski˘ı) and | X | ≤ c ( X ) χ ( X ) (Hajnal-Juh´asz).This solves a question that goes back to Bell, Ginsburg and Woods[6] and is mentioned in Hodel’s survey on Arhangel’ski˘ı’s Theorem[15]. In contrast to previous attempts we do not need any separa-tion axiom beyond T . Introduction
Two of the milestones in the theory of cardinal invariants in topologyare the following inequalities:
Theorem 1. (Arhangel’ski˘ı, 1969) [2] If X is a T space, then | X | ≤ L ( X ) χ ( X ) . Theorem 2. (Hajnal-Juh´asz, 1967) [13] If X is a T space, then | X | ≤ c ( X ) χ ( X ) . Here χ ( X ) denotes the character of X , c ( X ) denotes the cellularity of X , that is the supremum of the cardinalities of the pairwise disjointcollection of non-empty open subsets of X and L ( X ) denotes the Lin-del¨of degree of X , that is the smallest cardinal κ such that every opencover of X has a subcover of size at most κ .The intrinsic difference between the cellularity and the Lindel¨of de-gree makes it non-trivial to find a common extension of the two previousinequalities. The first attempt was done in 1978 by Bell, Ginsburg andWoods [6], who used the notion of weak Lindel¨of degree. The weakLindel¨of degree of X ( wL ( X )) is defined as the least cardinal κ such Mathematics Subject Classification.
Key words and phrases. cardinality bounds, cardinal invariants, cellularity, Lin-del¨of, weakly Lindel¨of, piecewise weakly Lindel¨of.The research that led to the present paper was partially supported by a grant ofthe group GNSAGA of INdAM. that every open cover of X has a ≤ κ -sized subcollection whose union isdense in X . Clearly, wL ( X ) ≤ L ( X ) and we also have wL ( X ) ≤ c ( X ),since every open cover without < κ -sized dense subcollections can berefined to a κ -sized pairwise disjoint family of non-empty open sets byan easy transfinite induction. Unfortunately, the Bell-Ginsburg-Woodsresult needs a separation axiom which is much stronger than Hausdorff. Theorem 3. [6] If X is a normal space, then | X | ≤ wL ( X ) χ ( X ) . It is still unknown whether this inequality is true for regular spaces,but in [6] it was shown that it may fail for Hausdorff spaces. Indeed,the authors constructed Hausdorff non-regular first-countable weaklyLindel¨of spaces of arbitrarily large cardinality.Arhangel’ski˘ı [3] got closer to obtaining a common generalization ofthese two fundamental results by introducing a relative version of theweak Lindel¨of degree, namely the cardinal invariant wL c ( X ), i.e. theleast cardinal κ such that for any closed set F and any family of opensets U satisfying F ⊆ S U there is a subcollection V ∈ [ U ] ≤ κ such that F ⊆ S V . Theorem 4. [3] If X is a regular space, then | X | ≤ wL c ( X ) χ ( X ) . O. Alas [1] showed that the previous inequality continues to holdfor Urysohn spaces, but it is still open whether it’s true for Hausdorffspaces.In [4] Arhangel’skii made another step ahead by introducing thenotion of strict quasi-Lindel¨of degree, which allowed him to give acommon refinement of the countable case of his 1969 theorem and theHajnal-Juh´asz inequality. He defined a space X to be strict quasi-Lindel¨of if for every closed subset F of X , for every open cover U of F and for every countable decomposition {U n : n < ω } of U thereare countable subfamilies V n ⊂ U n , for every n < ω such that F ⊂ S { S V n : n < ω } . It is easy to see that every Lindel¨of space is strictquasi-Lindel¨of and every ccc space is strict-quasi Lindel¨of. Arhangel’skiiproved that every strict quasi-Lindel¨of first-countable space has cardi-nality at most continuum.However, Arhangel’skii’s approach cannot be extended to higher car-dinals. Indeed, it’s not even clear whether | X | ≤ χ ( X ) is true for everystrict quasi-Lindel¨of space X . This inspired us to introduce the follow-ing cardinal invariants: Definition 5. • The piecewise weak Lindel¨of degree of X ( pwL ( X ) ) is definedas the minimum cardinal κ such that for every open cover U of COMMON EXTENSION 3 X and every decomposition {U i : i ∈ I } of U , there are ≤ κ -sized families V i ⊂ U i , for every i ∈ I such that X ⊂ S { S V i : i ∈ I } . • The piecewise weak Lindel¨of degree for closed sets of X ( pwL c ( X ) )is defined as the minimum cardinal κ such that for every closedset F ⊂ X , for every open family U covering F and for everydecomposition {U i : i ∈ I } of U , there are ≤ κ -sized subfamilies V i ⊂ U i such that F ⊂ S { S V i : i ∈ I } . As a corollary to our main result, we will obtain the following bound,which is the desired common extension of Arhangel’skii’s Theorem andthe Hajnal-Juh´asz inequality.
Theorem 6.
For every Hausdorff space X , | X | ≤ pwL c ( X ) · χ ( X ) . For undefined notions we refer to [11]. Our notation regarding car-dinal functions mostly follows [14]. To state our proofs in the mostelegant and compact way we use the language of elementary submod-els, which is well presented in [10].2.
A cardinal bound for the G δ -modification The following proposition collects a few simple general facts aboutthe piecewise weak Lindel¨of number which will be helpful in the proofof the main theorem.
Proposition 7.
For any space X we have: (1) pwL ( X ) ≤ pwL c ( X ) . (2) pwL c ( X ) ≤ L ( X ) . (3) pwL c ( X ) ≤ c ( X ) . (4) If X is T then wL c ( X ) ≤ pwL ( X ) .Proof. The first two items are trivial. To prove the third item, let F be a closed subset of X and V = S {V i : i ∈ I } be an open collectionsatisfying F ⊆ S V . Suppose c ( X ) ≤ κ . For every i ∈ I let C i bea maximal collection of pairwise disjoint non-empty open subsets of X such that for each C ∈ C i there is some V C ∈ V i with C ⊆ V C .By letting W i = { V C : C ∈ C i } , the maximality of C i implies that S V i ⊆ S W i and so F ⊆ S {∪W i : i ∈ I } . Since |W i | ≤ |C i | ≤ κ , wehave pwL c ( X ) ≤ κ .To prove the fourth item assume X is a regular space and let κ bea cardinal such that pwL ( X ) ≤ κ . Let F be a closed subset of X and U be an open cover of F . If U covers X we’re done. Otherwise useregularity to choose, for every p ∈ X \ S U an open set U p such that p ∈ U p and F ∩ U p = ∅ . Note that U ∪ { U p : p ∈ X \ F } is an open A. BELLA AND S. SPADARO cover of X , so by pwL ( X ) ≤ κ , there is a κ -sized subfamily V of U such that X ⊂ S V ∪ S { U p : p ∈ X \ F } . Hence F ⊂ S V and we aredone. (cid:3) Corollary 8. If X is a regular space then | X | ≤ pwL ( X ) · χ ( X ) .Proof. Combine Proposition 7, (4) and Arhangel’skii’s result that | X | ≤ wL c ( X ) · χ ( X ) for every regular space X . (cid:3) We state our main theorem in terms of the G κ -modification of aspace. Let κ be a cardinal number. By X κ we denote the topology on X generated by κ -sized intersections of open sets of X . We call X κ , the G κ -modification of X ; in case κ = ω we speak of the G δ -modificationof X and we often use the symbol X δ instead. This construction hasbeen extensively studied in the literature; various authors have triedto bound the cardinal functions of X κ in terms of their values on X (see, for example [8], [12], [16], [17], [18]) and results of this kind havefound applications to other topics in topology, like the estimation ofthe cardinality of compact homogeneous spaces (see [5], [8], [9] and[18]).By X cκ we denote the topology on X generated by G cκ -sets, that isthose subsets G of X such that there is a family { U α : α < κ } of opensets with G = T { U α : α < κ } = T { U α : α < κ } . In general, thetopology of X cκ is coarser than the G κ -modification of X , but if X is aregular space then X cκ = X κ . Theorem 9.
Let X be a Hausdorff space such that t ( X ) · pwL c ( X ) ≤ κ and X has a dense set of points of character ≤ κ . Then wL ( X cκ ) ≤ κ .Proof. Let F be a cover of X by G cκ -sets. Let θ be a large enoughregular cardinal and M be a κ -closed elementary submodel of H ( θ ) suchthat | M | = 2 κ and M contains everything we need (that is, X, F ∈ M , κ + 1 ⊂ M etc...).For every F ∈ F choose open sets { U α : α < κ } such that F = T { U α : α < κ } = T { U α : α < κ } . Claim 1.
F ∩ M covers X ∩ M . Proof of Claim 1.
Let x ∈ X ∩ M . Since F is a cover of X we can finda set F ∈ F such that x ∈ F . Moreover, using t ( X ) ≤ κ , we can find a κ -sized subset S of X ∩ M such that x ∈ S . Note that x ∈ U α ∩ S , forevery α < κ . Moreover, by κ -closedness of M , the set U α ∩ S belongsto M . Set B = T { U α ∩ S : α < κ } . Note that x ∈ B ⊂ F and B ∈ M .Therefore H ( θ ) | = ( ∃ G ∈ F )( x ∈ B ⊂ G ) and all the free variables inthe previous formula belong to M . Therefore, by elementarity we also COMMON EXTENSION 5 have that M | = ( ∃ G ∈ F )( x ∈ B ⊂ G ) and hence there exists a set G ∈ F ∩ M such that x ∈ G , which is what we wanted to prove. △ Claim 2.
F ∩ M has dense union in X . Proof of Claim 2.
Suppose by contradiction that X * S ( F ∩ M ). Thenwe can fix a point p ∈ X \ S ( F ∩ M ) such that χ ( p, X ) ≤ κ . Let { V α : α < κ } be a local base at p .For every F ∈ F ∩ M , let { U α ( F ) : α < κ } ∈ M be a sequence of opensets such that F = T { U α ( F ) : α < κ } = T { U α ( F ) : α < κ } . Notethat { U α ( F ) : α < κ } ⊂ M . Let C = { U α ( F ) : F ∈ F ∩ M, α < κ } .Note that C is an open cover of X ∩ M and C ⊂ M .For every x ∈ X ∩ M , we can choose, using Claim 1, a set F x ∈ F ∩ M such that x ∈ F x . Since p / ∈ F x , there is α < κ such that p / ∈ U α ( F x ).Hence we can find an ordinal β x < κ such that V β x ∩ U α ( F x ) = ∅ . Thisshows that U = { U ∈ C : ( ∃ β < κ )( U ∩ V β = ∅ ) } is an open cover of X ∩ M . Let U α = { U ∈ U : U ∩ V α = ∅} . Then {U α : α < κ } is adecomposition of U and hence we can find a κ -sized family V α ⊂ U α for every α < κ such that X ∩ M ⊂ S { S V α : α < κ } . Note thatby κ -closedness of M the sequence { S V α : α < κ } belongs to M andhence the previous formula implies that: M | = X ⊂ [ { [ V α : α < κ } So, by elementarity: H ( θ ) | = X ⊂ [ { [ V α : α < κ } But that is a contradiction, because p / ∈ S V α , for every α < κ . △ Since |F ∩ M | ≤ κ , Claim 2 proves that wL ( X cκ ) ≤ κ , as we wanted. (cid:3) As a first consequence, we derive the desired common extension ofArhangel’skii’s Theorem and the Hajnal-Juh´asz inequality.Recall that the closed pseudocharacter of the point x in X ( ψ c ( x, X ))is defined as the minimum cardinal κ such that there is a κ -sized family { U α : α < κ } of open neighbourhoods of x with T { U α : α < κ } = { x } .The closed pseudocharacter of X ( ψ c ( X )) is then defined as ψ c ( X ) =sup { ψ c ( x, X ) : x ∈ X } . Corollary 10.
Let X be a Hausdorff space. Then | X | ≤ pwL c ( X ) · χ ( X ) . A. BELLA AND S. SPADARO
Proof.
It suffices to note that in a Hausdorff space ψ c ( X ) · t ( X ) ≤ χ ( X )and hence if κ is a cardinal such that χ ( X ) ≤ κ then X cκ is a discreteset. Thus wL ( X cκ ) ≤ κ if and only if | X | = | X cκ | ≤ κ . (cid:3) Remark . Corollary 10 is a strict improvement of both Arhangel’skii’sTheorem and the Hajnal-Juh´asz inequality. Indeed, if S is the Sorgen-frey line and A ([0 , X = ( S × S ) ⊕ A ([0 , pwL c ( X ) = ℵ and L ( X ) = c ( X ) = c .Recall that a space is initially κ -compact if every open cover of car-dinality ≤ κ has a finite subcover (for κ = ω we obtain the usual notionof countable compactness). The following Lemma essentially says thatif X is an initially κ -compact spaces such that wL c ( X ) ≤ κ , then it sat-isfies the definition of pwL c ( X ) ≤ κ when restricted to decompositionsof cardinality at most κ . Lemma 11.
Let X be an initially κ -compact space such that wL c ( X ) ≤ κ and F be a closed subset of X . If U is an open cover of F and {U α : α < κ } is a κ -sized decomposition of U , then there are κ -sizedsubfamilies V α ⊂ U α such that F ⊂ S { S V α : α < κ } Proof.
Let U α = S U α . Then { U α : α < κ } is an open cover of F of cardinality κ , so by initial κ -compactness there is a finite subset S of κ such that F ⊂ { U α : α ∈ S } . Let now W = S {U α : α ∈ S } .We then have F ⊂ S W and hence by wL c ( X ) ≤ κ we can find a κ -sized subfamily W ′ of W such that F ⊂ S W ′ . Set now V α = { W ∈W ′ : W ∈ U α } . Then |V α | ≤ κ and F ⊂ S { S V α : α < κ } , as wewanted. (cid:3) Noticing that in the proof of Theorem 9 we only needed to apply thedefinition of pwL c ( X ) ≤ κ to decompositions of cardinality κ , Theorem9 and Lemma 11 imply the following corollaries. Corollary 12. [8]
Let X be an initially κ -compact space containing adense set of points of character ≤ κ and such that wL c ( X ) · t ( X ) ≤ κ .Then wL ( X cκ ) ≤ κ . Corollary 13. (Alas, [1] ) Let X be an initially κ -compact space with adense set of points of character κ , such that wL c ( X ) · t ( X ) · ψ c ( X ) ≤ κ .Then | X | ≤ κ . Open Questions
Corollary 8 can be slightly improved by replacing regularity with theUrysohn separation property (that is, every pair of distinct points can
COMMON EXTENSION 7 be separated by disjoint closed neighbourhoods). Indeed, in a similarway as in the proof of Proposition 7 (4) it can be shown that if X isUrysohn then wL θ ( X ) ≤ pwL ( X ), where wL θ ( X ) is the weak Lindel¨ofnumber for θ -closed sets (see [7]). Moreover, | X | ≤ wL θ ( X ) · χ ( X ) forevery Urysohn space X . However it’s not clear whether regularity canbe weakened to the Hausdorff separation property. That motivates thenext question. Question 3.1.
Is the inequality | X | ≤ pwL ( X ) · χ ( X ) true for every Haus-dorff space X ? Moreover, we were not able to find an example which distinguishescountable piecewise weak Lindel¨of number for closed sets from the strictquasi-Lindel¨of property.
Question 3.2.
Is there a strict quasi-Lindel¨of space X such that pwL c ( X ) > ℵ ? Finally, Arhangel’skii’s notion of a strict quasi-Lindel¨of space sug-gests a natural cardinal invariant. Define the strict quasi-Lindel¨of num-ber of X ( sqL ( X )) to be the least cardinal number κ , such that for everyclosed subset F of X , for every open cover U of F and for every κ -sized decomposition {U α : α < κ } of U there are κ -sized subfamilies V α ⊂ U α such that X ⊂ S { S V α : α < κ } . Obviously sqL ( X ) ≤ pwL c ( X ). It’snot at all clear from our argument whether the piecewise weak-Lindel¨ofnumber for closed sets can be replaced with the strict quasi-Lindel¨ofnumber in Corollary 10. Question 3.3.
Let X be a Hausdorff space. Is it true that | X | ≤ sqL ( X ) · χ ( X ) ? Even the following special case of the above question seems to beopen.
Question 3.4.
Let X be a strict quasi-Lindel¨of space. Is it true that | X | ≤ χ ( X ) ? References [1] O. T. Alas,
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