aa r X i v : . [ m a t h . GN ] N ov A NOTE ON RANK 2 DIAGONALS
ANGELO BELLA AND SANTI SPADARO
Abstract.
We solve two questions regarding spaces with a ( G δ )-diagonal of rank 2. One is a question of Basile, Bella and Ridderbosregarding weakly Lindel¨of spaces with a G δ -diagonal of rank 2 andthe other is a question of Arhangel’skii and Bella asking whetherevery space with a diagonal of rank 2 and cellularity continuumhas cardinality at most continuum. Introduction
A space is said to have a G δ -diagonal if its diagonal can be writtenas the intersection of a countable family of open subsets in the square.This notion is of central importance in metrization theory, ever sinceSneider’s 1945 theorem [14] stating that every compact Hausdorff spacewith a G δ diagonal is metrizable. Sneider’s result was later improvedby Chaber [8] who proved that every countably compact space with a G δ diagonal is compact and hence metrizable.Around the same time, Ginsburg and Woods [10] showed the influ-ence of G δ diagonals in the theory of cardinal invariants for topologicalspaces by proving that every space with a G δ diagonal without uncount-able closed discrete sets has cardinality at most continuum. Their resultled them to conjecture that every ccc space with a G δ diagonal musthave cardinality at most continuum. Shakhmatov [13] and Uspenskii[15] gave a pretty strong disproof to this conjecture by constructingTychonoff ccc spaces with a G δ -diagonal of arbitrarily large cardinal-ity. However, in the meanwhile, several strengthenings of the notionof a G δ -diagonal had been introduced, leading several researchers totest Ginsburg and Woods’s conjecture against these stronger diagonalproperties. That culminated in Buzyakova’s surprising result [7] thata ccc space with a regular G δ -diagonal has cardinality at most contin-uum. A space has a regular G δ -diagonal if there is a countable familyof neighbourhoods of the diagonal in the square such that the diagonalis the intersection of their closures. Mathematics Subject Classification.
Key words and phrases. cardinality bounds, weakly Lindel¨of, G δ - diagonal,neighbourhood assignment, dual properties. Another way of strengthening the property of having a G δ diagonalis by considering the notion of rank. Recall that given a family U ofsubsets of a topological space and a point x ∈ X , St ( x, U ) := S { U ∈U : x ∈ U } . The set St n ( x, U ) is defined by induction as follows: St ( x, U ) = St ( x, U ) and St n ( x, U ) = S { U ∈ U : U ∩ St n − ( x, U ) = ∅} for every n >
1. A space is said to have a diagonal of rank n if there isa sequence {U k : k < ω } of open covers of X such that T { St n ( x, U k ) : k < ω } = { x } , for every x ∈ X . By a well-known characterization,having a diagonal of rank 1 is equivalent to having a G δ -diagonal. Notealso that a space with a G δ -diagonal of rank 2 is necessarily T .Zenor [17] observed that every space with a diagonal of rank 3 alsohas a regular G δ -diagonal so by Buzyakova’s result, every ccc spacewith a diagonal of rank 3 has cardinality at most continuum. In [3],the first author proved the stronger result that every ccc space witha G δ -diagonal of rank 2 has cardinality at most 2 ω . The followingquestion is still open though: Question 1. (Arhangel’skii and Bella [1] ) Is every regular G δ -diagonalalways of rank 2? A positive answer would lead to a far-reaching generalization ofBuzyakova’s cardinal bound.Arhangel’skii and the first-named author proved in [1] that everyspace with a diagonal of rank 4 and cellularity ≤ c has cardinality atmost continuum, and leave open whether this is also true for spaceswith a diagonal of rank 2 or 3. Question 2.
Let X be a space with a diagonal of rank 2 or 3 andcellularity at most c . Is it true that | X | ≤ c . From Proposition 4.7 of [4] it follows that | X | ≤ c ( X ) ω for everyspace X with a diagonal of rank 3, which in turn that the answer toArhangel’skii and Bella’s question is yes for spaces with a diagonal ofrank 3. We show that the answer to their question is no for spaces witha diagonal of rank 2, by constructing a space with a diagonal of rank 2,cellularity ≤ c and cardinality larger than the continuum. That leadsto a complete solution to Arhangel’skii and Bella’s question.Recall that space X is weakly Lindel¨of provided that every opencover has a countable subfamily whose union is dense in X . This notionis a common generalisation of the Lindel¨of property and the countablechain condition (ccc). In view of the results by Ginsburg-Woods andBella mentioned above it is natural to consider the following question: Question 3. [4]
Let X be a weakly Lindel¨of space with a G δ -diagonalof rank . Is it true that | X | ≤ ω ? NOTE ON RANK 2 DIAGONALS 3
The above question was explicitly formulated in [4] and two partialpositive answers were obtained there under the assumptions that thespace is either Baire or has a rank 3 diagonal. Here we will prove thatQuestion 3 has a positive answer assuming that the space is normal.All undefined notions can be found in [12].2.
Spaces with a diagonal of rank 2
Recall that a neighbourhood assignment for a space X is a function φ from X to its topology such that x ∈ φ ( x ) for every x ∈ X . A set Y ⊆ X is a kernel for φ if X = S { φ ( y ) : y ∈ Y } . Following [11], wesay that a space X is dually P if every neighbourhood assignment in X has a kernel Y satisfying the property P . Of course, P implies dually P . A dually ccc space may fail to be even weakly Lindel¨of.Here we need the countable version of a well-known result of Erd¨osand Rado: Lemma 4.
Let X be a set with | X | > ω . If [ X ] = S { P n : n < ω } ,then there exist an uncountable set S ⊆ X and an integer n ∈ ω suchthat [ S ] ⊆ P n . Theorem 5. If X is a dually weakly Lindel¨of normal space with a G δ -diagonal of rank , then | X | ≤ ω .Proof. Let {U n : n < ω } be a sequence of open covers of X suchthat { x } = T { St ( x, U n ) : n < ω } for each x ∈ X . Assume bycontradiction that | X | > ω and for any n < ω put P n = {{ x, y } ∈ [ X ] : St ( x, U n ) ∩ St ( y, U n ) = ∅} . The assumption that the sequence {U n : n < ω } has rank 2 implies that [ X ] = S { P n : n < ω } . ByLemma 4 there exists an uncountable set S ⊆ X and an integer n such that [ S ] ⊆ P n . The collection { St ( x, U n ) : x ∈ S } consists ofpairwise disjoint open sets. From that it follows that, for any z ∈ X ,the set St ( z, U n ) cannot meet S in two distinct points, which impliesthat the set S is closed and discrete.We define a neighbourhood assignment φ for X as follows: if x ∈ S let φ ( x ) = St ( x, U n ) and if x ∈ X \ S let φ ( x ) = X \ S . Since X isdually weakly Lindel¨of, there exists a weakly Lindel¨of subspace Y suchthat X = S { φ ( y ) : y ∈ Y } . By the way φ is defined, it follows that S ⊆ S { φ ( y ) : y ∈ Y ∩ S } and hence S ⊆ Y . As X is normal, we maypick an open set V such that S ⊆ V and V ⊆ S { St ( x, U n ) : x ∈ S } .The trace on Y of the open cover { St ( x, U n ) : x ∈ X } ∪ { X \ V } witnesses the failure of the weak Lindel¨of property on Y . This is acontradiction and we are done. (cid:3) A. BELLA AND S. SPADARO
Related results for the classes of dually ccc spaces and for that ofcellular-Lindel¨of spaces were proved in [16] and [6].Finally we will construct a space with a diagonal of rank 2, cellularityat most continuum and cardinality larger than the continuum, thussolving Problem 2 from [1]. Recall that a κ -Suslin Line L is a continuouslinear order (endowed with the order topology) such that c ( L ) ≤ κ
Moore Machine (see, for example [9]) to (
S, τ ) we obtain a Moore space M ( S ) suchthat |M ( S ) | = | S | > ℵ and c ( M ( S )) ≤ ℵ . Recalling that Moorespaces have a diagonal of rank 2 (see Proposition 1.1 of [2]) and that c = ℵ under V = L , we see that X = M ( S ) satisfies the statement ofthe theorem. (cid:3) The above theorem also shows that the assumption that the spaceis Baire is essential in Proposition 4.5 from [4], thus solving a questionasked by the authors of [4] (see the paragraph after the proof of Lemma4.6).
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On spaces with regular G δ -diagonals , Pacific J. Math, (1972),959–963. Dipartimento di Matematica e Informatica, University of Catania,Citt`a Universitaria, Viale A. Doria 6, 95125 Catania, Italy
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