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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 33 - 37

F-points in countably compact spacesA. Bella and V. I. Malykhin�Abstract. Answering a question of A. V. Arhangel0ski��, weshow that any extremally disconnected subspace of a compactspace with countable tightness is discrete.2000 AMS Classi�cation: Primary 54A25; Secondary 54A10Keywords: F-point, extremal disconnectedness, compact, countably com-pact, pseudocompact, countable tightness.1. The resultsIn [1] Arhangel0ski�� noticed that under PFA any extremally disconnectedsubspace of a compact space with countable tightness is discrete. Then, heposed the natural question whether this result holds in ZFC.The aim of this short note is just to provide the full answer to such question.We actually manage to generalize the result by weakening the compactnessassumption.Henceforth, all space are assumed to be Tychono� spaces, unless otherwisespeci�ed.Recall that, for a given topological space X,the tightness at the point x 2 X,denoted by t(x;X), is the smallest cardinal � such that, whenever x 2 A � X,there exists a set B � A satisfying jBj � � and x 2 B.We say that x 2 X is a F-point if there are no disjoint open F�-sets U andV such that x 2 U \ V .It is evident that X is a F-space if and only if each x 2 X is a F-point in X.If Y is a subspace of the space X, we say that Y is countably compact inX provided that every in�nite subset of Y has an accumulation point in X.Of course, a space is countably compact if and only if it is countably compactin itself. If Y is countably compact in X, then Y is pseudocompact but notnecessarily countably compact.The next Lemma is essentially a combination of Theorem 1 in [2] and Propo-sition 3.1 in [3].�Work supported by the research project \Analisi Reale" of the Italian Ministero dell'Uni-versit�a e della Ricerca Scienti�ca e Tecnologica.



34 A. Bella and V. I. MalykhinLemma 1.1. Let X be a space, Y � X and y be a non-isolated point of Y . IfY is countably compact in X and t(y;X) � @0 then y is in the closure of somecountable discrete subspace of Y .Proof. Let A be a countable subset of Y n fyg such that y 2 A and �x afamily U = fUn : n 2 !g of closed neighbourhoods of y in X chosen so thatA \ TU = ?. Let S be the subset of X consisting of all points which are inthe closure of some countable set fzn : n 2 !g where zn 2 Un \ A for each n.Since Y is countably compact in X and y 2 Un \ A for each n, it easily followsthat y 2 S. As t(y;X) � @0, we may select a set fsn : n 2 !g � S so thaty 2 fsn : n 2 !g. Let fzin : n 2 !g be a sequence witnessing that si 2 S andput B = fzhj : 0 � h � j < !g. B is a subset of A which contains each si in itsclosure and so y 2 B. Furthermore, since B \TU = ? and B n Un is �nite forevery n, it follows that B is discrete. �Lemma 1.2. Let x be a F-point of the space X. If x is in the closure of somecountable discrete set N � X, then N [ fxg is homeomorphic to a subspace of�N.Proof. By contradiction, let us assume that there are two disjoint subset A;B �N such that x 2 A \ B. As N is discrete, for each y 2 N there is an openneighbourhood V (y) of y such that V (y) \ N = fyg. Let fai : i 2 !g andfbi : i 2 !g be enumerations of A and B. Let Pi be an open F�-set satisfyingai 2 Pi � V (ai) n SfV (bj) : j � ig and let Qi be an open F�-set satisfyingbi 2 Qi � V (bi) n SfV (aj) : j � ig. But now, letting P = SfPi : i 2 !g andQ = SfQi : i 2 !g, we get disjoint open F�-sets of X satisfying x 2 P \Q { incontrast with the fact that x is a F-point. �A crucial role is played here by the following result, which is a bit more generalversion than the one proved for countably compact spaces in [5], Corollary 4.Lemma 1.3. Let p 2 �N nN. If the subspace N [fpg is countably compact inthe space X, then t(p;X) > @0.Proof. Let us assume that N [ fpg is countably compact in the space X. Ofcourse, we may identify p with the trace in N of the family of all neighbourhoodsof p in X.Suppose �rst that p is a P-point. For any P 2 p, choose a partition ofP consisting of two in�nite sets P 0 and P 00 so that P 0 2 p and let x(P) be anaccumulation point of the set P 00 in X. The regularity of X guarantees that p isin the closure of the set A = fx(P) : P 2 pg � X. If X has countable tightnessat p, then we could �nd fPn : n < !g � p such that p 2 fx(Pn) : n < !g. But,since p is a P-point, there exists a neighbourhood U of p in X such that U \P 00nis �nite for each n < ! and so p =2 fx(Pn) : n < !g { a contradiction.If p is not a P-point, then there is a partition fAn : n < !g of N into in�nitesets such that for every n, An 62 p, and if P 2 p then jP \Anj = ! for in�nitely



F-points in countably compact spaces 35many n. Now put T = f [n<! Fn : Fn � An �nite;n < !g:For any T 2 T , select an accumulation point x(T) of T in X and let A =fx(T) : T 2 T g. Again, the regularity of X guarantees that p 2 A. If X hascountable tightness at p, then there exists a family fTn : n < !g � T such thatp 2 fx(Tn) : n < !g. For every n < !, let Tn = Sm<! Fnm, with Fnm a �nitesubset of Am for every m. PutK = F00 [ (F01 [ F11 ) [ (F02 [ F12 [ F22 ) [ : : : :Then K \An is �nite for every n, and as a consequence, P = N nK 2 p. Thus,taking a neighbourhood U of p in X so that U \ K = ?, we have that U \ Tnis �nite for every n. But this is a contradiction with p 2 fx(Tn) : n < !g. �Theorem 1.4. A non-isolated F-point in a countably compact space has notcountable tightness.Proof. Let x be a non-isolated F-point in the countably compact space X andassume that the tightness at x is countable. Thanks to Lemma 1.1, there isa countable discrete N � X such that x 2 N. By Lemma 1.2, we have thatN [ fxg is homeomorphic to a subspace of �N. But, according to Lemma 1.3,X has not countable tightness at x { a contradiction. �Corollary 1.5. A space containing a non-isolated F-point cannot be denselyembedded into a countably compact space with countable tightness.Proof. It is enough to observe that if Y is a dense estention of X, then a F-pointof X is still a F-point in the space Y . �Now, we easily get the answer to Arhangel0ski��'s question, in the followingmore general form:Corollary 1.6. An extremally disconnected subspace which is countably com-pact in a space with countable tightness is discrete.Observe that most of the results stated here are no longer true outside theclass of Tychono� spaces. Indeed, for any countable Hausdor� space X �x amaximal almost disjoint family A of in�nite closed discrete subsets of X anddenote by  (X) the set X [ A, topologized in such a way that X is an opensubspace of  (X) and a local base at A 2 A consists of the sets fAg [ (A n F),where F is a �nite subset of A. It is easy to see that  (X) is a Hausdor� spacewith countable tightness and X is countably compact in  (X). Now, if we takeas X a countable maximal space [4], then Lemma 1.1 fails for  (X), becauseno point of X is in the closure of a discrete subset of  (X). If we take as Xthe space N [ fpg, for some p 2 �N n N, then Lemma 1.3 fails for such  (X).Moreover, in both cases, X is a non-discrete extremally disconnected spacewhich is countably compact in a Hausdor� space with countable tightness, thatis a failure of Corollary 1.6.Coming back to Tychono� spaces, Corollary 1.6 may naturally suggest thefollowing:



36 A. Bella and V. I. MalykhinQuestion 1.7. Is it true that an extremally disconnected (dense) subspace of apseudocompact space with countable tightness is necessarely discrete?It is possible to show that in a pseudocompact space with character at most@1 every non-isolated point is in the closure of a discrete set. However, it is notclear whether Lemma 1.1 holds for pseudocompact spaces.Question 1.8. Does every non-isolated point of a pseudocompact space withcountable tightness belong to the closure of a discrete set?We �nish by observing that, at �rst glance, Corollary 1.6 could have beenbetter formulated for F-spaces, but this is only an impression.Indeed, we could consider the notion of point of extremal disconnectedness(ED-point), as a point x 2 X such that there are no disjoint open sets U;V � Xfor which x 2 U \ V . It is obvious that any ED-point is a F-point and the twonotions are in general di�erent, look, for instance, at the point !1 in the space!1 + 1. However, this is not the case for spaces of countable tightness:Proposition 1.9. Let x be a point of countable tightness in a Tychono� spaceX. If x 2 X is a F-point then x is also a ED-point.Proof. It is enough to observe that if U is an open set and x 2 U then thereexists an open F� set V � U such that x 2 V j U. For this, let A � U be acountable set such that x 2 A and for any a 2 A �x an open F� set Va satisfyinga 2 Va � U. It is clear that the set V = SfVa : a 2 Ag is what we are lookingfor. �References[1] A. V. Arhangel0ski��, Strong disconnectedness properties and remainders in compacti�ca-tions, Topology Appl. 107, 1-2 (2000), 3{12.[2] A. V. Arhangel0ski�� and A. Bella, Countable fan-tightness versus countable tightness,Comm. Math. Univ. Carolinae 37, 3 (1996), 565{576.[3] A. Bella and V. I. Malykhin, Tightness and resolvability, Comm. Math. Univ. Carolinae39, 1 (1998), 177{184.[4] E. K. van Dowen, Applications of maximal topologies, Topology Appl. 51 (1993), 125{140.[5] V. I. Malykhin, Countable spaces having no bicompacti�cations of countable tightness,Soviet Math. Dokl. 13 (1972), 1407{1411. Received October 2000Revised version April 2001
A. BellaDepartment of MathematicsUniversity of CataniaCitta' universitaria viale A. Doria 695125 Catania, Italy



F-points in countably compact spaces 37E-mail address: Bella@dipmat.unict.it
V. I. MalykhinState University of ManagementRjazanskij prospekt 99 Moscow109 542 RussiaE-mail address: slava@malykhin.mccme.ru