Weak extent, submetrizability and diagonal degrees

WEAK EXTENT, SUBMETRIZABILITY AND DIA GONAL DEGREES D. BASILE, A. BELLA, AND G. J. RIDDERBOS Abstract. W e show that if X has a zero-set diagonal and X 2 has coun table we ak exten t, then X is submet ri zable. This generalizes earlier results from Martin and Buzyak ov a. F urthermore we show that if X has a regular G δ - diagonal and X 2 has coun table weak exten t, then X condenses onto a secon d coun table Hausdorff space. W e also prov e several cardinality bounds inv olving v arious ty p es of diagonal degree. 1. Introduction A s pace is ca lle d submetrizable if it admits a coars er metr iz able top o logy . The diagonal of X 2 , denoted b y ∆ X , is the set { ( x, x ) : x ∈ X } . A space X is sa id to have a zero-set diagonal if there is a contin uous function f : X 2 → [0 , 1 ] such that ∆ X = f − 1 (0) and X is sa id to have a regular G δ -diagona l if ∆ X is a r egular G δ -subset of X , i.e. it is the intersection of co un tably many closed neig hbourho o ds. It is well-kno wn that every submetrizable space has a zero -set dia gonal, but the conv erse is fals e in g eneral (see the exa mple constructed in [15] and the re marks on it ma de in [2, Example 2.17]). This sugges ts to find conditions fo r a space with a zero-se t diagonal to b e submetriza ble. F or ex ample, in [13] H.W. Martin prov ed tha t separable spaces having a zer o-set diagonal are submetrizable. In another direction, in [7] R.Z. Buzyak ov a show ed that if X ha s a zer o-set diagonal and X 2 has countable extent then X is submetrizable. Separability and countable extent are indep e nden t pr op erties, but they hav e a quite natural co mmon weakening, namely countable weak e xten t. In the first part of our pap er, we giv e a s imultaneous genera lization of b oth the previous r esults by showing that spaces having a zer o-set dia gonal and whose squar e has co un table weak ex ten t are submetriza ble. Buzyak ov a also proved (see [7 , Theorem 2 .4 & 2.5 ]) that if X has a reg ular G δ - diagonal and either it is separa ble or X 2 has countable extent, then X condenses onto a s e cond-countable Hausdo rff space. Again, we give a simultaneous g eneral- ization of b oth these results by showing that if X 2 has c o unt able weak extent a nd a r egular G δ -diagona l, then X condenses onto a second-co untable Hausdorff space. In the seco nd par t of the pa pe r we will study cardinality b ounds on a s pace X according to the s p ecific way its diag onal is embedded in X 2 . Date : June 1, 2021. 2000 Mathematics Subje ct Classific ation. 54A25, 54C10, 54D20, 54E99. Key wor ds and phr ases. Submetrizable spaces, we ak exten t, regular G δ -diagonal, rank n - diagonal, weak Lindel¨ of num b er. 1 2 D. BASILE, A. BELLA, AND G. J. RIDDERBOS 2. Not a tion and terminology F or a ll undefined no tio ns we refer to [10]. Recall that X condenses o n to Y if there is a contin uous bijection from X onto Y . So a s pace is submetriza ble if and only if it condense s onto a metrizable space. The extent o f a space X , denoted b y e ( X ), is the supremum of the cardinalities of clos e d and discrete s ubsets of X . The weak extent of a space X , denoted b y we ( X ), is the least car dinal num b er κ such tha t for every op en cov er U of X there is a subset A of X of car dinality no g reater than κ such that St ( A, U ) = X . It is clea r that w e ( X ) ≤ d ( X ) and we ( X ) ≤ e ( X ). Note that spaces with co un table weak extent are calle d star countable by several a uthors (see, for insta nce [1]). F or a s pace X the weak-Lindel¨ of num b er of X , denoted by w L ( X ), is the lea s t cardinal κ such that every op en cov er of X has a s ubfamily of ca rdinality no greater than κ whose union is de ns e in X . Whenever B is a collection of subsets o f X a nd A ⊆ X , the star at A with resp ect to B , denoted by St( A, B ), is defined by the formula St( A, B ) = [ { B ∈ B : A ∩ B 6 = ∅ } . If we let St 0 ( A, B ) = A then, for n ∈ ω , the n -s tar around A is defined by induction: St n +1 ( A, B ) = St(St n ( A, B ) , B ) . Note that St 1 ( A, B ) = St( A, B ). If A = { a } we write St n ( a, B ) instea d of St n ( A, B ). If n ∈ ω , a nd κ is an infinite c ardinal, we sa y that a space X has a ra nk n G κ -diagona l (a strong rank n G κ -diagona l) if there is a sequence { U α : α < κ } of op en covers of X such that for all x 6 = y , there is some α < κ such that y 6∈ S t n ( x, U α ) ( y 6∈ S t n ( x, U α )). When κ = ω , we will simply write rank n -diagonal. W e will denote the minimal cardinal κ such that X has a rank n G κ -diagona l o r a str o ng r a nk n G κ -diagona l b y ∆ n ( X ) a nd s ∆ n ( X ), resp ectively . The fo r mula ∆ n ( X ) ≤ min { ∆ n +1 ( X ) , s ∆ n ( X ) } is obviously true. If n = 1 w e will o mit the nu mber 1. Recall that a space has a G δ -diagona l if and only if it has a rank 1-diag o nal (this was proved by Ceder in [9 , Lemma 5.4]). In analogy to Ceder’s r esult, Zenor prov ed in [17, Theorem 1] that a s pa ce X has a regular G δ -diagona l if a nd only if there is a sequence {U n : n ∈ ω } of op en covers o f X such that for a ll x 6 = y , there is a neighbo ur ho o d U o f x a nd some n ∈ ω such that y 6∈ S t ( U, U n ). In particular, if a spac e has a strong rank 2- diagonal, then it has a r egular G δ - diagonal. W e must say that a t present w e do not know an y exa mple of spa ces having a regular G δ -diagona l that do es not have a str ong rank 2 -diagona l. Even mo re int rig uing is the relationship b etw een regular G δ -diagona l and rank 2- diagonal. It is well-known that there exists a spa ce with a rank 2-diago nal that do es not hav e a regular G δ -diagona l, namely the Mr owk a spac e Ψ (see [2]). This easily follows from a result of McArthur ([14]), stating that a pseudo compac t s pa ce with a regular G δ -diagona l is metriza ble. But the following question from A. Bella ([4]) is still op en: Question 2 .1. Do es any sp ac e with a r e gular G δ -diagonal have a r ank 2 -diagonal? A go o d rea s on for ask ing such a questio n comes out fr om a compariso n of the following tw o facts. In [4] Bella prov ed that a cc c space with a rank 2- diagonal WEAK E XTENT, SUBMETRIZABILITY AND DIAGONAL DEGREES 3 has cardina lit y not exceeding 2 ω . Much more r ecently a nd with a certain effort, in [8] Buzyako v a has shown that a ccc space with a regular G δ -diagona l has a gain cardinality not exceeding 2 ω . Therefore, a p ositive answer to the previous question would imply a triv ial pr o of of the latter r e sult from the former. 3. Zero-set diagonal vs submetrizability The aim of this section is to provide a sim ultaneo us generalization of Martin and Buzyako v a’s res ults. The obvious wa y to acco mplish this is b y using the weak extent. How ever, we actually pre sent a formally strong er result o btained by means of an even weaker form of the weak extent of a square. The weak double extent of a spa ce X , denoted by w ee ( X ), is the smallest car dinal κ such that whenever U is an o p en cov er o f X 2 , there exists some A ⊆ X with | A | ≤ κ such that St( X × A, U ) = X 2 . The following is obvious. Prop ositio n 3 .1. F or any sp ac e X , we have w e ( X ) ≤ w ee ( X ) ≤ w e ( X 2 ) . By using Exa mple 3.3.4 in [16], w e are going to provide a spac e X such that we ( X ) < wee ( X ). Let Ψ b e the Mrowk a space A ∪ ω , wher e the cardina lity of A is c , and let Y b e the one- po in t co mpa ctification o f a discrete spa ce D of ca r dinality c . The spac e X = Ψ ⊕ Y is the to po logical sum of a separa ble spa ce a nd a compact space and so we have we ( X ) = ω . W rite A = { A α : α < c } and D = { d α : α < c } . Let U 1 = { Ψ × { d α } : α < c } , U 2 = { ( { A α } ∪ A α ) × ( Y \ { d α } ) : α < c } , U 3 = { { n } × Y : n < ω } , and finally U = U 1 ∪ U 2 ∪ U 3 ∪ { Y × Y } ∪ { Ψ × Ψ } ∪ { Y × Ψ } . Of course the family U is an o pen cover of X 2 . Assume that there exists a countable set C ⊆ X such that S t ( X × C, U ) = X 2 . This in tur n would imply the relation S t (Ψ × ( C ∩ Y ) , U 1 ∪ U 2 ∪ U 3 )) = Ψ × Y . Since we ha ve Ψ × Y \ ( S U 2 ∪ S U 3 ) ⊇ { ( A α , d α ) : α < c } , it sho uld b e { ( A α , d α ) : α < c } ⊆ S t (Ψ × ( C ∩ Y ) , U 1 ). But this would imply D ⊆ C ∩ Y , which is a contradiction. This suffices for the pro of that wee ( X ) > ω = w e ( X ). A further lo o k shows that we actua lly hav e w ee ( X ) = c . By rep ea ting the same constr uction, with the Ka tetov’s extensio n in place of Ψ and with D a set of cardinality 2 c , we get a Haus dorff space X such that we ( X ) = ω a nd we e ( X ) = 2 c . Right now, we do not have a space X fo r which w ee ( X ) < we ( X 2 ). Lemma 3. 2. If we e ( X ) = ω and F is a close d subset of X 2 and U is a c over of F by op en subsets of X 2 , t hen t her e is a c ountable subset A of X su ch that F ⊆ St( X × A, U ) . Theorem 3.3. If X has a zer o-set diagonal and w ee ( X ) = ω , then X is submetriz- able. Pr o of. Let f : X 2 → [0 , 1] be such that f − 1 (0) = ∆ X . Next, for n ∈ N we let C n = f − 1 ([ 1 / n , 1]). O f cour se C n is a closed subset of X 2 , and X 2 \ ∆ X = S n ∈ N C n . 4 D. BASILE, A. BELLA, AND G. J. RIDDERBOS F or n ∈ N , we let W n be defined by W n = { U × V : U × V ⊆ f − 1 (( 1 / 2 n , 1]) , V × V ⊆ f − 1 ([0 , 1 / 2 n )) & U, V op en in X } . Note that W n is a cov er of C n by op en subsets of X 2 . T o see this, fix n ∈ N and let ( x, y ) ∈ C n . W e have f ( x, y ) ∈ ( 1 / 2 n , 1], and therefo r e there exist op en subse ts U and V of X such that ( x, y ) ∈ U × V ⊆ f − 1 (( 1 / 2 n , 1]). Mo reov er, since ( y , y ) ∈ V × V and f ( y , y ) = 0 we can shrink V in such a way that V × V ⊆ f − 1 ([0 , 1 / 2 n )). Since we e ( X ) = ω , by the preceding lemma we may find a c o unt able subset B n of X such that C n ⊆ St( X × B n , W n ) . W e now let B = S n ∈ N B n , and we define F : X → [0 , 1] B by F ( x )( b ) = f ( x, b ) . W e will show that F is an injection. Since B is countable, this will imply that X is submetriza ble. Pick x, y ∈ X with x 6 = y . Then there is s ome n ∈ ω \ { 0 } with ( x, y ) ∈ C n . So we may find b ∈ B n and U × V ∈ W n such that ( x, y ) ∈ U × V and b ∈ V . The n ( x, b ) ∈ U × V and ( y , b ) ∈ V × V . F rom the definition of W n , it follows that f ( y , b ) < 1 / 2 n < f ( x, b ) , and there fo re F ( x ) 6 = F ( y ). This completes the pro o f.  The following is the anno unced generalizatio n of [13, Theorem 1] and [7, Theo rem 2.1]. Corollary 3.4. If X 2 has c ount able we ak ex t ent and a zer o-set diagonal , then X is su bmetrizable. In [7, Theore m 2.4 and 2 .5], R.Z. Buzyak ov a prov ed that if X ha s a reg ular G δ - diagonal and either it is separa ble or X 2 has countable extent, then X condenses onto a se cond-countable Hausdor ff space. F ollowing the same technique of Buzyako v a, we now gener alize those tw o results. Theorem 3.5. L et w e e ( X ) ≤ κ and assume t hat X has a r e gular G δ -diagonal. Then X c ondenses onto a Hau s dorff sp ac e of weight at most κ . Pr o of. Let ∆ X = T n<ω U n = T n<ω U n , and let C n = X 2 \ U n . W e define a family of op en se ts U as follows: U = { U × V : U × V ⊂ X \ U m , V × V ⊂ U m for s ome m ∈ ω & U, V op en in X } . Note that since ∆ X = T m ∈ ω U m , it follows that U is an op en cover of X 2 \ ∆ X . Since w ee ( X ) ≤ κ , we may find, for every n ∈ ω , a subset B n of X of car dinality at mos t κ such that C n ⊆ St( X × B n , U ) . If we let B = S n ∈ ω B n , then B is of cardinality at most κ and X 2 \ ∆ X ⊆ St( X × B , U ) . Now we let the family B consis t of all op en subse ts of X of one of the following forms: (1) { y : ( y , b ) ∈ U n } for so me b ∈ B a nd some n ∈ ω , (2) { x : ( x, b ) ∈ X 2 \ U n } for some b ∈ B and some n ∈ ω . WEAK E XTENT, SUBMETRIZABILITY AND DIAGONAL DEGREES 5 Then since | B | ≤ κ , we also hav e that |B | ≤ κ . W e will show that B is a Hausdor ff separating family (cf. [7]). So, pick p 6 = q . Then there is some b ∈ B and U × V ∈ U s uc h that b ∈ V a nd ( p, q ) ∈ U × V . Also, s ince U × V ∈ U , there is some m ∈ ω such that U × V ⊂ X \ U m & V × V ⊂ U m . This means that ( p, b ) ∈ U m and ( q , b ) ∈ X \ U m , and so we hav e p ∈ { y : ( y , b ) ∈ U m } q ∈ { x : ( x, b ) ∈ X 2 \ U m } , and since these op en se ts ar e disjoint members of B , this shows that B is Hausdor ff separating.  Corollary 3.6. If X 2 has c ountable we ak extent and a r e gular G δ -diagonal, then X c ondenses onto a se c ond c ountable Hausdorff s p ac e. 4. S ome cardinal inequalities In this sectio n we prov e v arious cardinality bo unds involving differe nt types of di- agonal degr ee. W e star t off by showing that for Hausdor ff spaces X the inequalities | X | ≤ 2 d ( X ) s ∆( X ) and | X | ≤ w e ( X ) ∆ 2 ( X ) hold. Next, we shall prov e that if X is either a Bair e s pa ce with a rank 2- diagonal or a space with a rank 3-diag onal, then its car dinality is bo unded by w L ( X ) ω . W e do not know if the sa me inequality is still true fo r spa ces having a stro ng r ank 2-diago nal. How ever, w e can prove that, for such spaces, the inequality | X | ≤ wL ( X ) π χ ( X ) holds. Finally , we will show that the last formula is true for homog eneous s paces having a reg ular G δ -diagona l. Prop ositio n 4 .1. F or any Hausdorff sp ac e X we have | X | ≤ 2 d ( X ) s ∆( X ) . Pr o of. Let κ = d ( X ) s ∆( X ) and fix a family {U α : α < κ } that witnesses the fact that X has a strong r ank 1 G κ -diagona l. Let D b e a dense subset of X of cardinality at mos t κ . W e define a map F : X → P ( D ) κ by F ( x )( α ) = D ∩ St( x, U α ) . W e only hav e to show that this map is one-to-o ne. First of all, note that since D is dense, we always hav e x ∈ F ( x )( α ) . Now let x 6 = y . The n we may find α < κ with y 6∈ St( x, U α ). B ut then, s ince F ( x )( α ) ⊆ St( x, U α ), it follows that y 6∈ F ( x )( α ). So a s y ∈ F ( y )( α ), it follows that F ( x )( α ) 6 = F ( y )( α ).  One could try to co njecture the b o und 2 d ( X )∆( X ) , but the Ka tetov extension of the discrete space ω dispr oves it. It is s eparable, it ha s a G δ -diagona l and its cardinality is 2 c . T aking into a ccount a r esult of Gins burg a nd W o o ds, se e [11, Theor em 9.4 ], which states that if X is a T 1 space, then its cardinality is b ounded by 2 e ( X )∆ ( X ) , it is quite na tural to w onder whether the prev io us pr o po sition can b e improved as follows: Question 4 .2. Is the c ar dinality of a Hausdorff sp ac e X b oun de d by 2 we ( X ) s ∆( X ) ? 6 D. BASILE, A. BELLA, AND G. J. RIDDERBOS If, in the previous question, we replace s ∆( X ) with ∆ 2 ( X ), we ca n actually prov e the following strong er b ound. Prop ositio n 4 .3. F or any Hausdorff sp ac e X we have | X | ≤ w e ( X ) ∆ 2 ( X ) . Pr o of. Let κ = w e ( X ) and λ = ∆ 2 ( X ). Fix a seq ue nce of op en covers {U α : α < λ } witnessing the fact that X has a rank 2 G λ -diagona l. F or every α < λ , we may fix a subs et A α of X with | A α | ≤ κ s uch that X = St( A α , U α ). W e let A = S α<λ A α . Note that | A | ≤ κ · λ . W e may fix a map f : X → A λ with the prop erty tha t for x ∈ X and α < λ we have that f ( x )( α ) = a ∈ A α and x ∈ St( a, U α ). T o c omplete the pro of we will show that such a mapping is injective. So fix x 6 = y . Then we may find α < λ such that St( x, U α ) ∩ St( y , U α ) = ∅ . Now let p = f ( x )( α ). Then x ∈ St ( p, U α ), and so als o p ∈ St( x, U α ). This means that p 6∈ St( y , U α ) a nd ther efore y 6∈ St( p, U α ). This implies that p 6 = f ( y )( α ). So the mapping f is injective a nd this completes the pro of.  This result sho uld be compared with the inequality | X | ≤ we ( X ) psw ( X ) , obtained by R. Ho del (see [3] for a n alter native and dir ect pro o f; see also [1 2]). The K atetov extension of ω witnesses that in the las t tw o formulas it is no t p ossible to put ∆( X ) at the exp onent. How ever, one may s till try to conjecture to impr ov e Ginsbur g- W oo ds’ inequality by moving down e ( X ) from the exp onent. This question was already published b y Be lla in 1996 (see [6 ]), but we think is worthy to rep ea t it here. Question 4.4 . Do es the ine quality | X | ≤ e ( X ) ∆( X ) hold for any T 1 sp ac e X ? In [4, Theorem 2 ], Bella pr ov ed that the cardina lit y of a Hausdorff spa c e X is b ounded b y 2 c ( X )∆ 2 ( X ) . This was done by an a pplication of the Er d¨ os-Rado Theorem. F or Ba ire spaces with a ra nk 2-diago nal this bound can b e considera bly improv ed. Prop ositio n 4 .5. If X a Bair e sp ac e with a r ank 2-diagonal then, | X | ≤ w L ( X ) ω . Pr o of. This follows from P rop osition 4.3, the fact that we ( X ) ≤ d ( X ) a nd the following lemma.  Lemma 4.6. If X is a Bair e sp ac e with a G δ -diagonal then, d ( X ) ≤ wL ( X ) ω . Pr o of. Let w L ( X ) = κ and let {U n : n < ω } be a sequence of op en covers of X witnessing the fact that X has a rank 1-diago nal. F or every n < ω , we fix a family V n ⊆ U n of ca rdinality κ whose union is dense in X . Next we let V = S n<ω V n and D n = S V n . Then |V | ≤ κ , and D n is an o pe n and de ns e subset of X for every n . Since X is a Baire spac e , this means that D = T n<ω D n is a dense subset of X . So to co mplete the pr o of it suffices to show that | D | ≤ κ ω . WEAK E XTENT, SUBMETRIZABILITY AND DIAGONAL DEGREES 7 W e fix s ome well-ordering on V and we define a map f : D → V ω as fo llows f ( d )( n ) = min { V ∈ V : d ∈ V ∈ V n } . W e will show that f is a n injection. So fix x, y ∈ D with x 6 = y . Then y 6∈ St( x, U n ) for some n ∈ ω . Let V = f ( x )( n ). Then x ∈ V a nd s ince V n is a refinement of U n , this mea ns that V ⊆ St( x, U n ). So we hav e that y 6∈ V and therefor e f ( x )( n ) 6 = f ( y )( n ). This completes the pro of.  W e could a sk whether the Ba ire assumption in Pro po sition 4.5 is necessar y . This is an op en ques tion, but we c a n prove that for spa ces having a ra nk 3- diagonal the following is tr ue. Prop ositio n 4 .7. If X has a r ank 3 - diagonal then, | X | ≤ w L ( X ) ω . Pr o of. Let w L ( X ) = κ and let {U n : n < ω } be a sequence of op en covers of X witnessing the fact that X has a rank 3-diago nal. F or every n < ω , we fix a family V n ⊆ U n of cardinality κ whose union is dense in X . Next we let V = S n<ω V n . Of co urse we hav e |V | ≤ w L ( X ). Note tha t whenever U ∈ U n , there is s ome V ∈ V n such that U ∩ V 6 = ∅ . So it follows that for every x ∈ X and n ∈ ω , there is some V ∈ V n such that St( x, U n ) ∩ V 6 = ∅ . Also note that in this c a se V ⊆ St 2 ( x, U n ). W e fix a well-ordering o n V a nd we de fine a map F : X → V ω as fo llows F ( x )( n ) = min { V ∈ V : V ∈ V n & St( x, U n ) ∩ V 6 = ∅ } . W e hav e just shown that F is well-defined. It remains to show that F is an injection. So let x, y ∈ X with x 6 = y . By a ssumption, there is some n ∈ ω such that St 2 ( x, U n ) ∩ St( y , U n ) = ∅ . Since F ( x )( n ) ⊆ St 2 ( x, U n ) and F ( y )( n ) ∩ St ( y , U n ) 6 = ∅ , it follows that F ( x )( n ) 6 = F ( y )( n ). This shows that F is an injection a nd this completes the pro of.  The discrete cellular it y of a spa ce X is the cardinal num b er dc ( X ) = s up { |U | : U is a discrete family of op en subsets of X } . T he last result should b e compared with the inequality | X | ≤ 2 dc ( X )∆ 3 ( X ) prov ed in [5]. Note that, at least for regular spaces, w e have dc ( X ) ≤ w L ( X ) and the gap ca n b e artitrar ely larg e . W e do not know if the last tw o men tioned inequa lities are true fo r spa ces with a str o ng rank 2-diago nal. Question 4.8 . L et X b e a sp ac e with a st ro ng r ank 2-diagonal. Is it the c ase that • | X | ≤ w L ( X ) ω ? • | X | ≤ 2 dc ( X ) ? How ever, for spaces of countable π -character, we hav e the answer. Prop ositio n 4 .9. L et X b e a sp ac e with a str ong r ank 2-diagonal. Then | X | ≤ w L ( X ) π χ ( X ) . Pr o of. Let {U n : n < ω } b e a sequence of op en cov ers of X witnessing the fact that X ha s a s trong ra nk 2-diagonal and let κ = π χ ( X ) and λ = wL ( X ). F or every x ∈ X , w e let V x = { V ( x, α ) : α < κ } b e a lo cal π - base a t x . F or n < ω , we fix a family W n ⊆ U n of cardinality λ w ho se unio n is dense in X . 8 D. BASILE, A. BELLA, AND G. J. RIDDERBOS Next we let W = S n<ω W n . Note that |W | ≤ λ . Since U n is a cover of X , it follows that whenever V is a non-empt y o pen subset of X , then V ∩ W 6 = ∅ fo r some W ∈ W n . W e fix a well-ordering on W and we define a map F : X → W κ × ω as fo llows, F ( x )( α, n ) =  ∅ , if V ( x, α ) 6⊆ St ( x, U n ) , min { W ∈ W n : W ∩ V ( x, α ) 6 = ∅ } , otherwise . By the remar ks ma de b efore, the ma p F is well-defined. F or x ∈ X and n < ω , we let W ( x , n ) b e defined by W ( x , n ) = [ { F ( x )( α, n ) : α ∈ κ } . Note that by definition of F , we hav e that W ( x, n ) ⊆ St (St( x, U n ) , W n ) and since W n is a refinement of U n , it follows that W ( x , n ) ⊆ St 2 ( x, U n ) . Claim. x ∈ W ( x , n ) for every n ∈ ω . Proof of Claim. T o see this, let O x be an op en neighbourho o d o f x . Then V ( x, α ) ⊆ O x ∩ St( x, U n ) for some α < κ . By definition of F , it follows that F ( x )( α, n ) ∩ V ( x, α ) 6 = ∅ and therefor e F ( x )( α, n ) ∩ O x 6 = ∅ . Since F ( x )( α, n ) ⊆ W ( x , n ), it follows that x ∈ W ( x , n ) and this pr ov es the claim. ◭ So for every x ∈ X , we hav e tha t { x } ⊆ \ n<ω W ( x , n ) ⊆ \ n<ω St 2 ( x, U n ) = { x } . This shows tha t F is an injection and this completes the pro of.  F o r ho mo geneous spa ces, the pr evious pro po sition ca n b e improved. Note that if X is homog eneous and π χ ( X ) = κ , then there is a co llection { V ( x, α ) : x ∈ X , α < κ } o f non-empty open subsets o f X suc h that for e very x ∈ X , V x = { V ( x, α ) : α < κ } is a lo ca l π -base at x and whenever O x a nd O y a re op en neighbo urho o ds o f x and y res p ectively , there is some α < κ s uch that V ( x, α ) ⊆ O x and V ( y , α ) ⊆ O y . F o r example, if p ∈ X is fixed and { V α : α < κ } is a lo cal π -base at p in X , then we may define V ( x, α ) = h x [ V α ], where h x is a homeomor phis m of X mapping p onto x . Prop ositio n 4. 10. L et X b e a homo gene ous sp ac e with a r e gular G δ -diagonal. Then | X | ≤ w L ( X ) π χ ( X ) . Pr o of. Fix a sequence {U n : n < ω } o f op en covers of X witnessing the fact that X has a reg ular G δ -diagona l. F ur thermore, let π χ ( X ) = κ and w L ( X ) = λ and fix a collectio n { V ( x, α ) : x ∈ X , α < κ } o f no n-empt y o pe n s ubsets of X with the prop erty sta ted just b efor e this pro po sition. Next, for n < ω , we fix a family W n ⊆ U n of cardinality λ w ho se unio n is dense in X . Note that since U n is a cov er of X , if follows that whenever V is a non-empty op en subset o f X , then V ∩ W 6 = ∅ for some W ∈ W n . W e let W = S n<ω W n and we fix a well-ordering on W . Note that |W | ≤ w L ( X ). WEAK E XTENT, SUBMETRIZABILITY AND DIAGONAL DEGREES 9 W e now define a map F : X → W ω × κ as follows, F ( x )( n, α ) = min { W ∈ W : W ∈ W n & W ∩ V ( x, α ) 6 = ∅} . W e hav e just show ed that F is well-defined. It remains to verify that F is an injection, so let x, y ∈ X with x 6 = y . Then there is some n < ω and op en neighbourho o ds O x and O y of x and y r esp ectively such that St( Ox, U n ) ∩ O y = ∅ . By the pr op erty of our lo ca l π -ba s es, it follows that there is some α < κ such that V ( x, α ) ⊆ O x and V ( y , α ) ⊆ O y . Now re c a ll that W n is a refinement o f U n , and therefor e , since V ( x, α ) ⊆ Ox , w e hav e the following: F ( x )( n, α ) ⊆ St( O x, U n ) . F urther more, b y construction we have that F ( y )( n, α ) ∩ O y 6 = ∅ s o it follows that F ( x )( n, α ) 6 = F ( y )( n, α ). This shows that F is an injection and this completes the pro of.  References [1] O. T. A las, L. R. Junqueira, and R. G. Wil s on, Countability and star c overing pr op erti es , T opol ogy and its A pplications 1 58 (2011), no. 4, 620–626. [2] A. V. Arhangel ′ skii and R. Z. Buzy ako v a, The r ank of t he diagonal and submetrizability , Commen t. M ath. Univ. Carolin. 47 (2006), no. 4, 585–597. [3] D. Basile and A. 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Zenor, On sp ac es with r e gular G δ -diagonals , Pacific J. Math. 4 0 (1972), 759–763. 10 D. BASILE, A. BELLA, AND G. J. RIDDERBOS Universit ` a degli Studi di Ca t a nia, Dip ar timento di Ma tema tica e In forma tica, Viale Andrea Doria 6, 95125 Ca t ania, It al y E-mail addr ess : basile@dmi.uni ct.it Universit ` a degli Studi di Ca t a nia, Dip ar timento di Ma tema tica e In forma tica, Viale Andrea Doria 6, 95125 Ca t ania, It al y E-mail addr ess : bella@dmi.unic t.it F acul ty of Electrical Eng ineering, Ma themat ics and Computer S cience, TU Delf t, Postbus 5031 , 260 0 GA Delft, the Netherlands E-mail addr ess : G.F.Ridderbos@ tudelft.nl URL : http://aw. twi.tudelft.nl/~ridderbos