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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 13 - 28

An operation on topological spacesA. V. Arhangel0ski��Abstract. A (binary) product operation on a topologicalspace X is considered. The only restrictions are that some el-ement e of X is a left and a right identity with respect to thismultiplication, and that certain natural continuity requirementsare satis�ed. The operation is called diagonalization (of X). Twoproblems are considered: 1. When a topological space X admitssuch an operation, that is, when X is diagonalizable? 2. Whatare necessary conditions for diagonalizablity of a space (at a givenpoint)? A progress is made in the article on both questions. Inparticular, it is shown that certain deep results about the topo-logical structure of compact topological groups can be extendedto diagonalizable compact spaces. The notion of a Moscow spaceis instrumental in our study.2000 AMS Classi�cation: 54D50, 54D60, 54C35Keywords: C-embedding, Diagonalizable space, Hewitt-Nachbin completion,Moscow space, pseudocompact space, separability, Stone-�Cech compacti�ca-tion, tightness 1. Diagonalizable spacesIn this article we build upon some ideas and techniques from [5], showing thatthey are applicable in a much more general setting. The key new idea is mate-rialized below in a new notion of a diagonalizable space, which turns out to bea very broad generalization of the notion of a semitopological semigroup withidentity. It also generalizes the notion of a Mal0tsev space. Diagonalizablityis preserved by retracts and by products. Thus, a diagonalizable space neednot be homogeneous. Moreover, every zero-dimensional �rst countable space isdiagonalizable. However, despite its very general nature, diagonalizablity turnsout to be so strong a property, that we are able to extend some importanttheorems about compact topological groups to compact diagonalizable spaces.These results involve Stone-�Cech compacti�cations, C-embeddings, and prod-ucts; in particular, they extend the classical results of I. Glicksberg [10] andE. van Douwen [8] (see also [12]). A central role in what follows also belongs



14 A.V. Arhangel0ski��to the notion of a Moscow space, which was recently shown to have delicateapplications in topological algebra.A topological space X will be called diagonalizable at a point e 2 X if thereexists a mapping � of the square X � X in X satisfying the following twoconditions:1) �(x;e) = �(e;x), for each x 2 X;2) For each a 2 X, the mappings �a and �a of the space X into itself,de�ned by the formulas �a(x) = �(x;a) and �a(x) = �(a;x) for eachx 2 X, are continuous at x = e.The mapping � in this case is called a diagonalizing mapping (at e), or a diag-onalization of X at e, and the mappings �a and �a are called, respectively, theright action and the left action by a on X, corresponding to the product oper-ation �. If in the de�nition above the mapping � can be chosen to be jointlycontinuous at (e;a) and (a;e) for each a 2 X, we say that X is continuouslydiagonalizable at e. Clearly, every space X is diagonalizable at every isolatedpoint of X. If X is (continuously) diagonalizable at every point e 2 X, then Xis called (continuously) diagonalizable.A space X with a �xed separately (jointly) continuous mapping � : X�X !X and a �xed point e 2 X will be called a semitopoid (a topoid) with identitye if � is a diagonalization of X at e, that is, if �(x;e) = �(e;x) = x, for eachx 2 X. The next assertion is obvious.Proposition 1.1. If a space X is (continuously) diagonalizable at some pointa of X, and X is homogeneous, then X is (continuously) diagonalizable.Example 1.2.1) Every topological, and even every paratopological, group G is contin-uously diagonalizable: as a continuous diagonalization mapping � atthe neutral element e of G we can take just the product operation:�(x;y) = xy, for each (x;y) in G�G. It remains to refer to homogene-ity of G.Therefore, Sorgenfrey line is continuously diagonalizable, since it is aparatopological group.2) Every semitopological group G, that is, a group G with a topology suchthat the product operation in G is separately continuous (with respectto each argument), is diagonalizable at the neutral element e by theproduct operation. Since every semitopological group G is a homoge-neous space, it follows from Proposition 1.1 that every semitopologicalgroup is diagonalizable.3) Let X be a Mal0tsev space, that is, a space with a continuous mappingf of the cube X � X � X in X such that f(x;y;y) = f(y;y;x) = x forall x and y in X (such f is called a continuous antimixer on X). ThenX is continuously diagonalizable. Indeed, �x any e in X, and de�ne amapping � of X � X in X by the formula:�(x;y) = f(x;e;y):



An operation on topological spaces 15Clearly, � is continuous, and �(e;x) = f(e;e;x) = x = f(x;e;e) =�(x;e). Thus, � is a continuous diagonalization mapping at e, and Xis continuously diagonalizable.4) Similarly, every space with a separately continuous antimixer is diago-nalizable.5) Every retract of a topological group is a Mal0'tsev space (see [13]).Therefore, every retract of a topological group is continuously diago-nalizable. In particular, every absolute retract is continuously diago-nalizable. Hence, every Tychono� cube is continuously diagonalizable.Notice that a Mal0tsev space, unlike a topological group, need not behomogeneous (consider, for example, the closed unit interval). Thus,diagonalizable spaces do not have to be homogeneous.Several simple statements below demonstrate that the class of di-agonalizable spaces is much larger than the classes of semitopologicalgroups or Mal0tsev spaces.Proposition 1.3. Every linearly ordered topological space X with the smallestelement e is continuously diagonalizable at e.Proof. Let < be a linear ordering on X, generating the topology of X, suchthat e is the smallest element of X. For arbitrary (x;y) 2 X �X, put �(x;y) =maxfx;yg. Clearly, � is a continuous diagonalizing mapping at e. �Theorem 1.4. Every linearly ordered compact space X is continuously diago-nalizable at least at one point.Proof. Indeed, every compact space X, the topology of which is generated bya linear ordering <, has the smallest element with respect to < [9]. �Corollary 1.5. Every homogeneous linearly ordered compact space is continu-ously diagonalizable.Proof. This follows from Theorem 1.4 and Proposition 1.1. �Example 1.6. The \double arrow" space is continuously diagonalizable, sinceit is compact, homogeneous, and linearly ordered.The conclusion in Theorem 1.4 can be considerably strengthened if we assumethat the topology of X is generated by a well ordering. Indeed, we have thefollowingTheorem 1.7. If X is a topological space, the topology of which is generatedby a well ordering <, then X is continuously diagonalizable.Proof. Assume that e is any point of X. Put Y = fx 2 X : x � eg andZ = fx 2 X : e < xg. Then Y and Z are open and closed subsets of X, thespace Y is linearly ordered, and e is the last element of Y . From Proposition1.3 it follows that Y is continuously diagonalizable at e (consider the reverseordering of Y ).It remains to apply the next Lemma:



16 A.V. Arhangel0ski��Lemma 1.8. Assume that Y is an open and closed subspace of X, e 2 Y , and Yis (continuously) diagonalizable at e. Then X is (continuously) diagonalizableat e.Proof. Put Z = X nY . The sets Y �Y , Y �Z, Z �Y , and Z �Z are pairwisedisjoint and open and closed in X � X. Together they cover X � X.Let us �x a (continuous) diagonalizing mapping  for the space Y at e. Thenwe de�ne a diagonalizing mapping � for X at e as follows.If (x;y) 2 Y � Y , we put �(x;y) =  (x;y).If (x;y) 2 Y � Z, we put �(x;y) = y.If (x;y) 2 Z � Y , we put �(x;y) = x.If (x;y) 2 Z � Z, we put �(x;y) = e.Clearly, � is a (continuous) diagonalizing mapping for X at e. �Let us call a space X continuously homogeneous if there exist a point e 2 Xand a continuous mapping h of X into the space Hp(X) of homeomorphisms ofX onto itself in the topology of pointwise convergence satisfying the followingconditions:(1) hx(e) = x, for each x 2 X, where hx = h(x); and(2) he = h(e) is the identity mapping of X onto itself.Such a mapping h will be called a homogeneity Cp-structure on X at e.Clearly, every continuously homogeneous space is homogeneous.Proposition 1.9. Every continuously homogeneous space X is diagonalizable.Proof. Let h : X ! Hp(X) be a homogeneity Cp-structure on X at a pointe 2 X. Put �(x;y) = hx(y), for each (x;y) 2 X � X, where hx = h(x). Itis easily veri�ed that the mapping � is separately continuous. We also have:�(e;x) = he(x) = x, since he is the identity mapping, and �(x;e) = hx(e) = x,by the other property of homogeneity Cp-structure. Thus, X is diagonalizableat e. Since X is homogeneous, it follows that X is diagonalizable. �Theorem 1.10. Every retract of a (continuously) diagonalizable space is (con-tinuously) diagonalizable.Proof. Assume that X is a (continuously) diagonalizable space, Y a subspaceof X, and r a retraction of X onto Y . Take any point e in Y , and �x adiagonalizing mapping � : X � X ! X at e.De�ne a mapping �r of Y � Y in Y by the formula �r = r�(y;z), for ev-ery (y;z) in Y � Y . Clearly, if � is (separately) continuous, then �r is also(separately) continuous.Take any y 2 Y . Then�r(y;e) = r�(y;e) = r(y) = yand �r(e;y) = r�(e;y) = r(y) = y;since y 2 Y and r is a retraction of X onto Y . Therefore, Y is (continuously)diagonalizable at e. �



An operation on topological spaces 17Remark 1.11. Notice, that a space X is (continuously) diagonalizable at e 2 Xif and only if there exists a (continuous) separately continuous mapping � ofthe product space X � X onto the diagonal �X = f(x;x) : x 2 Xg such that�(x;e) = �(e;x) = (x;x), for each x in X. This obvious observation explainsthe name \diagonalizable space".There is another curious result on diagonalizablity involving retractions. Ob-serve that for any e 2 X the subspaces feg � X and X � feg are retracts ofX �X (under the obvious projections). Now let us ask the following question:when the subspace (X � feg) [ (feg � X) is a retract of X � X? If this is thecase, we will call the space X crosslike at e 2 X. If (X � feg) [ (feg � X) is aretract of X �X under a separately continuous retraction, then X will be saidto be weakly crosslike (at e). For example, the closed unit interval and the realline are crosslike spaces.Proposition 1.12. If a space X is weakly crosslike at e 2 X, then X is diag-onalizable at e.Proof. Fix a separately continuous retraction r of X � X onto the subspace(X � feg) [ (feg � X). For each x 2 X, put f(e;x) = f(x;e) = x. Then f isa continuous mapping of (X � feg) [ (feg � X) onto X. Clearly, the mapping� = f � r is a diagonalization of X at e. �Similarly, the next assertion is proved:Proposition 1.13. If a space X is crosslike at e 2 X, then X is continuouslydiagonalizable at e.A space X is called zero-dimensional at a point e 2 X if there exists a baseof X at e consisting of open and closed sets (notation: ind(e;X) = 0).Theorem 1.14. If a space X is zero-dimensional at a point e 2 X, and e is aG� in X, then X is crosslike at e, and, hence, continuously diagonalizable at e.Proof. We can �x a countable family fVn : n 2 !g of open and closed neighbor-hoods of e in X such that Vn+1 � Vn, for each n 2 !, and feg = \fVn : n 2 !g.Put Wn = Vn � (X n Vn+1), Un = (X n Vn) � Vn, W = [fWn : n 2 !g, andU = [fUn : n 2 !g. Obviously, the sets U and W are open in X � X, and(feg � (X n feg) � W , (X n feg) � feg) � U.It is easy to check that U and W are disjoint, and that they are closed in(X � X) n f(e;e)g. Therefore, the set K = (X � X) n (U [ W [ feg) is openin X � X. For each (x;y) 2 W , put r(x;y) = (e;y). For each (x;y) 2 U, putr(x;y) = (x;e). For each (x;y) 2 (X � X) n (U [ W), put r(x;y) = (e;e).Clearly, r is a continuous retraction of X � X onto the cross (X � feg) [(feg � X) at e. Hence, X is crosslike at e, and, by Proposition 1.13, X iscontinuously diagonalizable at e. �The same idea leads to one more elementary result in the same direction.Recall that, for a non-empty space X, the equality Ind(X) = 0 signi�es thatfor any two disjoint closed subsets P and F in X there exists an open andclosed subset W such that P � W and F \ W = ? (see [9]).



18 A.V. Arhangel0ski��Proposition 1.15. Let X be a space and e a point in X such that the subspaceZ = (X � X) n f(e;e)g of X � X satis�es the condition Ind(Z) = 0. Then thespace X is crosslike at e.Proof. Put P = feg � X and F = X � feg. Then P and F are disjoint closedsubsets of Z. Since Ind(Z) = 0, there exists an open and closed subset W of Zsuch that P � W and F \ W = ?.Now we de�ne a mapping r of X �X in (feg �X) [(X �feg) as follows. If(x;y) 2 W , let r(x;y) = (e;y). If (x;y) 2 Z n W , let r(x;y) = (x;e). Finally,we put r(e;e) = (e;e). Clearly, the restriction of r to W is continuous, since itis the restriction of the projection mapping of X �X. Similarly, the restrictionof r to Z nW is continuous. Therefore, r is continuous at all points of Z. SinceZ is open in X � X, to see that the mapping r is continuous, we only have tocheck its continuity at the point (e;e). However, the continuity of r at (e;e) isalso obvious.Finally, we observe that r is the identity mapping on (feg �X) [(X �feg).Thus, (feg � X) [ (X � feg) is a retract of X � X. �Theorem 1.14 shows how much more general is the diagonalizablity assump-tion, than the assumption that the space has a (separately) continuous antim-ixer. Indeed, according to [15], every compact space with a separately con-tinuous antimixer is a Dugundji compactum, and it is well known that every�rst countable Dugundji compactum is metrizable. We also see from Theorem1.14 that diagonalizablity of a compact space does not impose any homogeneityrestrictions on the space. In that the diagonalizablity di�ers drastically fromthe assumptions that X is a paratopological group or a semitopological group.Example 1.16. Let X be a space, e a point of X, and Pe(X) the space ofall closed subsets of X containing e, in the Vietoris topology. Put Z = Pe(X),and de�ne a mapping � : Z � Z ! Z by the rule: �(A;B) = A [ B, for any(A;B) 2 Z � Z.It is easily veri�ed that the mapping � is continuous. It is also clear that�(E;A) = A = �(A;E), for each A 2 Z, where E = feg. Therefore, � is acontinuous diagonalization of the space Z = Pe(X) at the point E = feg 2Pe(X).Since there is no reason to believe that the space Pe should be diagonalizableat every point, the above conclusion suggests that the space Pe normally canbe expected to be not homogeneous and provides some means for proving that.Though the proof of the next statement is obvious, the result itself is quiteimportant.Proposition 1.17. The product of any family of (continuously) diagonalizablespaces is a continuously diagonalizable space.Similar assertion holds for crosslike spaces. In conclusion of this section, wemention a curious corollary of Theorem 1.10.



An operation on topological spaces 19Theorem 1.18. Suppose X is a space such that X � Y is homeomorphic to a(continuously) diagonalizable space, for some space Y . Then X is also (contin-uously) diagonalizable.Proof. Indeed, X is a retract of X �Y . It remains to apply Theorem 1.10. �2. Some necessary conditions for diagonalizablityA space X is called Moscow at a point e 2 X if, for every open set U theclosure of which contains e, there exists a G�-subset P of X such that e 2 P � U(see [1, 5]). If X is Moscow at every point, we call X a Moscow space.A space X is called weakly Klebanov at a point e 2 X if for every family 
 ofG�-subsets of X such that the closure of [
 contains e there exists a G�-subsetP of X such that e 2 P � [
. We say that X is weakly Klebanov if X isweakly Klebanov at every point of X. Clearly, every weakly Klebanov space isMoscow, and every space of countable pseudocharacter is weakly Klebanov.The importance of the notion of Moscow space comes from the role it playsin connection with C-embeddings; see about that [6] and [4]. Besides, a non-trivial result on Moscow spaces is the theorem that every Dugundji compactumis Moscow (see [15]); it follows that every compact (actually, every pseudocom-pact) topological group is a Moscow space.The simplest example of a non-Moscow space is the one-point (Alexandro�)compacti�cation of an uncountable discrete space. Note that the tightnessof this space is countable. On the other hand, it was shown in [5] that everytopological (and even semitopological) group of countable tightness is a Moscowspace. This again underlines the signi�cance of the concept of a Moscow spacefor topological algebra.One of our main results is the next theorem:Theorem 2.1. Suppose X is a space of countable tightness diagonalizable (ate 2 X). Then X is weakly Klebanov (at e).Proof. Let A be a subset of X which is the union of a family of G�-subsets ofX, and e any point in A. We have to show that there exists a G�-subset P suchthat e 2 P � A.Since the tightness of X is countable, there exists a countable subset B of Asuch that e 2 B. For each b 2 B we �x a G�-set Pb such that b 2 Pb � A.Let us also �x a diagonalizing mapping � of X�X into X at e. For each b 2 Bconsider the mapping �b of X into X given by the formula: �b(x) = �(x;b), forevery x 2 X.Since � is a diagonalizing mapping at e, �b is continuous at e. Therefore,��1(Pb) contains a G�-set Mb such that e 2 Mb, since �b(e) = �(e;b) = b. Thenthe set F = \fMb : b 2 Bg is also a G�-set in X, and e 2 F .Take any point a 2 F . We have �b(a) 2 Pb � A, for each b 2 B, sinceF � ��1b (Pb). Thus, �(a;b) = �b(a) 2 A, for each b 2 B. However, e 2 B,and the function �(a;x) is continuous with respect to the second argument atx = e. It follows that �(a;e) 2 A. Since � is a diagonalizing mapping at e, wehave �(a;e) = a. Therefore, a 2 A, that is, F � A. The proof is complete. �



20 A.V. Arhangel0ski��The assumption that the tightness of X is countable can be considerablyweakened but can not be completely removed. The �-tightness of a space X ata point e 2 X is said to be countable [5] if for each open subset U such that eis in the closure of U there exists a countable subset B of U such that e 2 B(notation: t�(e;X) � !). If the �-tightness of X is countable at every pointe 2 X, we say that the �-tightness of X is countable, and write t�(X) � !.Introducing a mild, obvious, change in the proof of Theorem 2.1, we obtaina proof of the next statement:Theorem 2.2. Every diagonalizable (at a point e) space X of countable �-tightness is Moscow (at e).It is worth noting that for every dyadic compactum X the �-tightness ofX is countable, while if the tightness of a dyadic compactum X is countable,then X is metrizable (see [7]). In particular, the �-tightness of every Tychono�cube is countable. This shows that the countability of �-tightness is much,much weaker restriction than the countability of tightness. However, the nextexample shows that we can not completely drop it.Example 2.3. The space of ordinals !1 + 1 is continuously diagonalizable, byTheorem 1.7. Nevertheless, this space is easily seen to be not Moscow [4].Of course, this happens because the �-tightness of !1 + 1 is not countable(precisely at the point !1). Observe that the space !1 + 1 does not admit aseparately continuous antimixer, since it is compact but not dyadic. Observealso that, by Theorem 1.17, the space (!1 + 1)� is continuously diagonalizable,for every cardinal number �.Example 2.4. Let � be an uncountable cardinal number and A� the one-point(Alexandro�) compacti�cation of a discrete space of cardinality �. Then A� isnot diagonalizable (at the unique non-isolated point of A�). Obviously, A� isa Fr�echet-Urysohn space; hence, the tightness of A� is countable. Assume nowthat A� is diagonalizable. Then, by Theorem 2.1, A� is Moscow, a contradic-tion. It follows that A� is not diagonalizable.Note that the usual convergent sequence is continuously diagonalizable byTheorem 1.7 or by Theorem 1.14. Note also, that the space A� is compact,zero-dimensional, Hausdor�, and satis�es the �rst axiom of countability at allpoints except one, the non-isolated point. Thus, Theorem 1.14 can not be muchimproved.It is well known that a compact topological group of countable tightnessis metrizable (see [7]). For diagonalizable compact spaces we have a parallelstatement with a weaker conclusion. Recall that a compact space X is said tobe !-monolithic if, for every countable subset A of X, the closure of A in X isa space with a countable base.Theorem 2.5. Every diagonalizable !-monolithic compact Hausdor� space Xof countable tightness is �rst countable.Proof. Take any point x 2 X. Since X is compact Hausdor�, it is enoughto show that x is a G�-point in X. The space X is Fr�echet-Urysohn, and X



An operation on topological spaces 21is �rst countable at a dense set Y of points (since every !-monolithic compactHausdor� space of countable tightness has these properties [3]). Therefore,there exists a sequence fyn : n 2 !g of points of Y converging to x. On theother hand, X is weakly Klebanov, by Theorem 2.1.It remains to apply the following obvious lemma:Lemma 2.6. Suppose X is a weakly Klebanov space and fyn : n 2 !g is asequence of G�-points in X converging to x. Then x is also a G�-point in X.Note that the space A� in Example 2.4 is an !-monolithic compact Hausdor�space of countable tightness. Theorem 2.5 clari�es, why it is not diagonalizable:it is because it is not �rst countable.Corollary 2.7. Every diagonalizable Corson compactum is �rst countable.Proof. Indeed, every Corson compact space is monolithic and Fr�echet-Urysohn(see [3]). It remains to apply Theorem 2.5. �In the next result we assume the Continuum Hypothesis (CH). It is notclear whether the statement remains true without this assumption.Theorem 2.8. (CH) Every diagonalizable sequential compact Hausdor� spaceX is �rst countable.Proof. Let Y be the set of all points of X at which X satis�es the �rst axiom ofcountability. Then Y is G�-dense in X, by a theorem in [2] (here we use (CH)).On the other hand, from Theorem 2.1 it follows that X is weakly Klebanov.Assume now that X 6= Y . Then Y is not closed in X, since Y is dense in X.Therefore, since X is sequential, there exists a point x 2 X n Y and a sequencefyn : n 2 !g of points of Y converging to x. It follows from Lemma 2.6 that xis a G�-point in X. Since X is compact Hausdor�, we conclude that X is �rstcountable at x. This is a contradiction with x =2 Y and de�nition of Y . �3. Diagonalizablity and C-embeddingsIn this section, we combine our results on diagonalizablity and a result ofM.G. Tkachenko to obtain several new results on C-embeddings and Stone-�Cechcompacti�cations. For more results on C-embeddings in the context of topo-logical groups see [11]. Here is Uspenskij's 's modi�cation [15] of Tkachenko'sresult from [14]:Theorem 3.1 (Tkachenko). If X is a Moscow Tychono� space, then everyG�-dense subspace Y of X is C-embedded in X.Theorem 3.2. Let X be a compact diagonalizable space of countable �-tight-ness. Then X is the Stone- �Cech compacti�cation of any G�-dense subspace Yof X.Proof. Indeed, X is a Moscow space, by Theorem 2.1. Therefore, by Theorem3.1, Y is C-embedded in X. It follows that Y is pseudocompact and X =�Y . �



22 A.V. Arhangel0ski��The next statement is a typical application of Theorem 2.2.Theorem 3.3. If a Tychono� space X is diagonalizable at e 2 X, and the�-tightness of X at e is countable, then either e is a G�-point in X, or thesubspace Y = X n feg is C-embedded in X.Proof. Assume that e is not a G�-point in X. Then Y is G�-dense in X. ByTheorem 2.2, X is Moscow at e. Since Y is G�-dense in X, it follows, by anobvious modi�cation of Theorem 3.1 (see [4]), that Y is C-embedded in X. �Corollary 3.4. Assume that a Tychono� space X is diagonalizable at e 2 X,the �-tightness of X at e is countable, and the subspace Y = X nfeg is Hewitt-Nachbin complete. Then e is a G�-point in X.Proof. Assume that e is not isolated in X. Then, since Y = X n feg is Hewitt-Nachbin complete, Y is not C-embedded in X. Now it follows from Theorem3.3 that e is a G�-point in X. �Corollary 3.5. Assume that X is a pseudocompact Tychono� space diagonal-izable at a point e 2 X such that the �-tightness of X at e is countable. Theneither X is �rst countable at e, or the subspace X n feg is pseudocompact.Proof. Since ever pseudocompact Tychono� space is �rst countable at everyG�-point, it follows from Theorem 3.3 that the subspace Y = X n feg is C-embedded in X. Therefore, since X is pseudocompact, the space Y must bepseudocompact as well. �Here is a result in the same direction, in which the assumption on X doesnot contain explicitly a restriction on the tightness of X.Theorem 3.6. Assume that X is a pseudocompact Tychono� space diagonal-izable at a point e 2 X. Assume also that the next condition is satis�ed: (�)For each open subset U of X such that e 2 U n U, the subspace U n feg is notpseudocompact. Then X is �rst countable at e.Proof. Clearly, we can assume that the point e is not isolated in X. Thencondition � implies that the subspace X nfeg is not pseudocompact. It followsfrom Corollary 3.5 that, to complete the proof, it remains to show that the�-tightness of X at e is countable.Take any open set U such that e 2 U nU. By (�), the subspace Z = U nfeg isnot pseudocompact. Therefore, there exists a discrete family � = fVn : n 2 !gof non-empty open subsets in Z. However, the subspace U is pseudocompact,since X is pseudocompact and U is a canonical closed subset of X (see [9]).It follows that the sequence (Vn : n 2 !) converges to e. Clearly, Vn \ U isnon-empty, for each n 2 !. Choosing a point xn 2 Vn \ U for each n 2 !, weobtain a sequence of points of the set U converging to e. Hence, the �-tightnessof X at e is countable. �The condition (�) in Theorem 3.6 may look a little arti�cial. However, thereare several natural corollaries of Theorem 3.6. Recall that a subset A of a spaceX is called locally closed if A = B\C, where B is a closed subset of X and C is



An operation on topological spaces 23an open subset of X. The next three statements follow directly from Theorem3.6.Corollary 3.7. Assume that X is a pseudocompact Tychono� space diagonal-izable at a point e 2 X such that every locally closed pseudocompact subspaceof X is closed in X. Then X is �rst countable at e.Corollary 3.8. Assume that X is a pseudocompact Tychono� space diagonaliz-able at a point e 2 X and such that the subspace Xnfeg is Dieudonn�e complete.Then X is �rst countable at e.Corollary 3.9. Assume that X is a pseudocompact Tychono� space diagonal-izable at a point e 2 X and such that the subspace X n feg is metacompact.Then X is �rst countable at e.The list of corollaries to Theorem 3.6 can be easily expanded.4. Diagonalizable separable spacesThe results obtained in the preceding sections are, in particular, applicableto separable spaces. We present several such applications below.Theorem 4.1. If a separable space X is diagonalizable at e 2 X, then X isMoscow at e.Proof. This statement is a direct corollary of Theorem 2.1 and the obvious factthat the �-tightness of every separable space is countable. �A space X is called a G�-extension of a space Y if Y is a G�-dense subspaceof X. A space X may have many di�erent G�-extensions. For example, everycompacti�cation of a pseudocompact Tychono� space X is a G�-extension ofX, and usually there are many such compacti�cations.However, it turns out that few of these extensions should be expected to bediagonalizable. This is demonstrated by the next "uniqueness" result.Theorem 4.2. If a Tychono� space X is a G�-extension of a separable spaceY , and X is diagonalizable and Hewitt-Nachbin complete, then X is the Hewitt-Nachbin completion �Y of Y .Proof. The space X is also separable. Therefore, by Theorem 3.1, X is Moscow.Since Y is G�-dense in X, it follows Theorem 3.1 that Y is C-embedded inX. Therefore, since X is Hewitt-Nachbin complete, X is the Hewitt-Nachbincompletion of X. �With the help of Theorem 4.2, we could easily construct many further ex-amples of non-diagonalizable separable spaces.The notion of diagonalizablity can be also applied to show that G�-extensionsof spaces, in general, should not be expected to be homogeneous. This is basedon the following key lemma from [6]:Lemma 4.3. If a Tychono� space X is a homogeneous G�-extension of aMoscow space Y , then X is also a Moscow space and Y is C-embedded inX.



24 A.V. Arhangel0ski��Theorem 4.4. If a Tychono� space X is a homogeneous G�-extension of aseparable diagonalizable space Y , and X is Hewitt-Nachbin complete, then X isthe Hewitt-Nachbin completion �Y of Y .Proof. By Theorem 4.1, the space Y is Moscow. Since X is homogeneous andY is G�-dense in X, it follows from Lemma 4.3 that X is also Moscow and Y isC-embedded in X. Since X is Hewitt-Nachbin complete, we can conclude thatX is the Hewitt-Nachbin completion �Y of Y . �Corollary 4.5. If X is a compact Hausdor� homogeneous extension of a sep-arable pseudocompact diagonalizable space Y , then X is the Stone- �Cech com-pacti�cation of Y .Proof. Indeed, Y is G�-dense in X, since Y is pseudocompact, and X is Hewitt-Nachbin complete, since X is compact Hausdor�. It remains to apply Theorem4.2. �The next statement is proved by a similar argument.Corollary 4.6. If X is a Hausdor� compacti�cation of a separable pseudocom-pact space Y , and X is diagonalizable, then X is the Stone- �Cech compacti�ca-tion of Y .We know that every zero-dimensional Hausdor� space of countable pseu-docharacter is diagonalizable. We also established several conditions underwhich diagonalizable spaces are Moscow or even have countable pseudocharac-ter. Since the class of Moscow spaces is an extension of the class of spaces ofcountable pseudocharacter, it is natural to ask if every zero-dimensional Moscowspace is diagonalizable. Theorem 4.1 is instrumental in �nding a compact coun-terexample.Example 4.7. Let �! be the Stone-�Cech compacti�cation of the discrete space!, and e 2 �! n !. Let us show that �! is not diagonalizable at e.Assume the contrary. Then the space Z = �! � �! is, obviously, diagonal-izable at the point (e;e). Since the space �! is separable, the space Z is alsoseparable. Now it follows from Theorem 4.1 that Z is Moscow at the point(e;e). However, this is not the case, as it was shown in [5]. Thus, not everycompact Moscow space of countable �-tightness is diagonalizable.The next two results are related in an obvious way to the classical theoremsin [10] and [12] (see also [6] and [4]).Theorem 4.8. Assume that Y� is a separable Tychono� space with a diag-onalizable Hewitt-Nachbin complete G�-extension X�, for each � 2 A, wherejAj � 2!. Then the next formula holds for the Hewitt-Nachbin extensions �Y�:�f�Y� : � 2 Ag = ��fY� : � 2 Ag:Proof. By Theorem 4.2, �Y� = X�, for each � 2 A. Therefore, �f�Y� : � 2 Agis a G�-extension of the space �fY� : � 2 Ag. Obviously, �f�Y� : � 2 Ag isHewitt-Nachbin complete. Applying again Theorem 4.2 and Proposition 1.17,we conclude that �f�Y� : � 2 Ag = ��fY� : � 2 Ag. �



An operation on topological spaces 25Corollary 4.9. Assume that Y� is a separable pseudocompact space with adiagonalizable Hausdor� compacti�cation bY�, for each � 2 A, where jAj � 2!.Then the next formula holds for the Stone- �Cech compacti�cations �Y�:�f�Y� : � 2 Ag = ��fY� : � 2 Ag:Proof. To deduce this statement from Theorem 4.8, it is enough to observe thatevery pseudocompact space is G�-dense in each Hausdor� compacti�cation of it,and that the Hewitt-Nachbin completion of any pseudocompact space coincideswith the Stone-�Cech compacti�cation of Y . �We conclude this section with the next obvious corollary of Theorem 4.1.Corollary 4.10. If a separable Tychono� space X is diagonalizable at a pointe 2 X, and X n feg is Hewitt-Nachbin complete, then e is a G�-point in X.5. Continuously diagonalizable spacesFollowing M.G. Tkachenko [14], we say that the o-tightness of a space X at apoint e 2 X is countable (and write ot(e;X) � !) if, for each family 
 of opensubsets of X such that e 2 [
, there exists a countable subfamily � of 
 suchthat e 2 [�. If this is true for every point e in X, we say that the o-tightnessof X is countable.Theorem 5.1. If a space X is continuously diagonalizable at a point e 2 X,and the o-tightness of X at e is countable, then X is Moscow at e.Proof. Let U be any open subset of X such that e is in the closure of U.Obviously, we may assume that e is not in U.Let � be the family of all open subsets W of X such that, for some openneighborhood OW of e (which we now �x), xy 2 U for each x 2 OW and eachy 2 W (that is, �(OW � W) � U). Then � is a base of the space U, since theoperation � is jointly continuous at (e;x), for each x 2 X.Therefore, e 2 [�. Since the o-tightness of X is countable, it follows thatthere exists a countable subfamily � of � such that e is in the closure of [�.Put G = [� and P = \fOW : W 2 �g. Then P is a G�-set in X, since � iscountable, and e 2 P , e 2 G.Take any a 2 P . We want to show that a 2 U. We may assume that a is note, since e 2 U. Then, for each W 2 �, a 2 OW which implies that aW � U.Therefore, aG � U. Since ax depends continuously on the second argument xat x = e, and e 2 G, it follows that ae 2 aG � U. Finally, since ae = a, weobtain: a 2 U, that is, e 2 P � U, and X is a Moscow space. �Corollary 5.2. If a space X is continuously diagonalizable at a point e 2 X,and the Souslin number of X is countable, then X is a Moscow space.Proof. It is enough to observe that if the Souslin number of X is countable,then the o-tightness of X is also countable [14]. �Corollary 5.3. If X is a continuously diagonalizable Tychono� space with thecountable Souslin number, then every G�-dense subspace Y of X is C-embeddedin X.



26 A.V. Arhangel0ski��Proof. The space X is Moscow, by Theorem 5.1. It follows from Theorem 3.1that every G�-dense subspace Y of X is C-embedded in X. The next statementis a typical application of Theorem 2.2. �Corollary 5.4. If a Tychono� space X is continuously diagonalizable at e 2 X,and the o-tightness of X at e is countable, then either e is a G�-point in X, orthe subspace Y = X n feg is C-embedded in X.Proof. Assume that e is not a G�-point in X. Then Y is G�-dense in X. ByTheorem 5.1, X is Moscow at e. Since Y is G�-dense in X, it follows, byTheorem 3.1, that Y is C-embedded in X. �Corollary 5.5. Suppose a Tychono� space X is continuously diagonalizable ate 2 X, the o-tightness of X at e is countable, and the subspace Y = X n feg isHewitt-Nachbin complete. Then e is a G�-point in X.Proof. Assume that e is not isolated in X. Then, since Y = X n feg is Hewitt-Nachbin complete, Y is not C-embedded in X. Now it follows from 5.4 that eis a G�-point in X. �The next result should be compared to 3.5Corollary 5.6. Assume that X is a pseudocompact Tychono� space contin-uously diagonalizable at a point e 2 X such that the o-tightness of X at e iscountable. Then either X is �rst countable at e, or the subspace X n feg ispseudocompact.Proof. Since every pseudocompact Tychono� space is �rst countable at everyG�-point, from Corollary 5.4 it follows that the subspace Y = X n feg is C-embedded in X. Therefore, since X is pseudocompact, the space Y must bepseudocompact as well. �Many results, proved in the previous section for separable diagonalizablespaces, have their counterparts for continuously diagonalizable spaces with thecountable Souslin number. Their proofs do not di�er much, so we just formulatea few such results below, omitting the proofs.Theorem 5.7. Assume that a Tychono� space X is a G�-extension of a spaceY such that the Souslin number of Y is countable, and assume also that Xis continuously diagonalizable and Hewitt-Nachbin complete. Then X is theHewitt-Nachbin completion �Y of Y .Theorem 5.8. If a Tychono� space X is a homogeneous G�-extension of acontinuously diagonalizable space Y with the countable Souslin number, and Xis Hewitt-Nachbin complete, then X is the Hewitt-Nachbin completion �Y ofY .Corollary 5.9. If X is a compact Hausdor� homogeneous extension of a pseu-docompact continuously diagonalizable space Y with the countable Souslin num-ber, then X is the Stone- �Cech compacti�cation of Y .



An operation on topological spaces 27Corollary 5.10. Assume that Y� is a separable pseudocompact space with acontinuously diagonalizable Hausdor� compacti�cation bY�, for each � 2 A.Then the next formula holds for the Stone- �Cech compacti�cations �Y�:�f�Y� : � 2 Ag = ��fY� : � 2 Ag:In connection with Corollaries 5.9 and 5.10, see [8] and [10].References[1] Arhangel0ski�� A.V., Functional tightness, Q-spaces, and �-embeddings, Comment. Math.Univ. Carol. 24:1 (1983), 105{120.[2] Arhangel0ski�� A.V., On bicompacta hereditarily satisfying Souslin's condition. Tightnessand free sequences. Soviet Math. Dokl. 12 (1971), 1253{1257.[3] Arhangel0ski�� A.V., Topological Function Spaces. Kluwer Academic Publishers, 1992.[4] Arhangel0ski�� A.V., Topological groups and C-embeddings. Submitted, 1999.[5] Arhangel0ski�� A.V., On a Theorem of W.W. Comfort and K.A. Ross, Comment. Math.Univ. Carolinae 40:1 (1999), 133{151.[6] Arhangel0ski�� A.V., Moscow spaces, Pestov{Tkachenko Problem, and C-embeddings. Toappear in CMUC.[7] Arhangel0ski�� A.V. and V.I. Ponomarev, Fundamentals of General Topology in Problemsand Exercises. (D. Reidel Publ. Co., Dordrecht-Boston, Mass., 1984).[8] van Douwen E., Homogeneity of �G if G is a topological group. Colloquium Mathematicum41 (1979), 193{199.[9] Engelking R., General Topology. PWN, Warszawa, 1977.[10] Glicksberg I., Stone- �Cech compacti�cations of products, Trans. Amer. Math. Soc. 90(1959), 369{382.[11] Hernandez S., Sanchis M., and M.G. Tkachenko, Bounded sets in spaces and topologicalgroups. Topology and Appl. 101:1 (2000), 21{44.[12] Hu�sek M., The Hewitt realcompacti�cation of a product. Comment. Math. Univ. Carol.11 (1970), 393{395.[13] Reznichenko E.A., V.V. Uspenskij, Pseudocompact Mal0tsev spaces. Topology and Appl.86 (1998), 83{104.[14] Tkachenko M.G., The notion of o-tightness and C-embedded subspaces of products, Topol-ogy and Appl. 15 (1983), 93{98.[15] Uspenskij V.V., Topological groups and Dugundji spaces, Matem. Sb. 180:8 (1989), 1092{1118. Received March 2000
Arhangel0ski��, A.V.(January 1-June 15):Department of MathematicsMorton Hall 321Ohio University,Athens, Ohio 45701USAE-mail address: arhangel@bing.math.ohiou.edu



28 A.V. Arhangel0ski��(June 15{December 30):Kutuzovskij Prospect, h. 33 apt. 137Moscow 121165RussiaE-mail address: arhala@arhala.mccme.ru