532 A. Bella, I.V. Yaschenko General remarks on AP spaces In ëptë, it was observed that!1 + 1 is not AP. The rst author showed in ëb1ë that every scat

Transcription

1 Comment.Math.Univ.Carolinae 40,3 è1999è531í On AP and WAP spaces A. Bella, I.V. Yaschenko Abstract. Several remarks on the properties of approximation by points èapè and weak approximation by points èwapè are presented. We look in particular at their behavior in product and at their relationships with radiality, pseudoradiality and related concepts. For instance, relevant facts are: èaè There is in ZFC a product of a countable WAP space with a convergent sequence which fails to be WAP. èbè C p over ç-compact space is AP. Therefore AP does not imply even pseudoradiality in function spaces, while it implies Frçechet-Urysohn property in compact spaces. ècè WAP and AP do not coincide in C p. Keywords: AP space, WAP space, pseudoradial space, radial space, product, compact space, submaximal space, function space Classication: 54A25, 54D55 All spaces under consideration are assumed to be Tychono. For all undened notions see ëenë. The following denitions were originated in Categorical Topology. Denition 0.1 èëptëè. A space X is said to have the property of Approximation by Points èweak Approximation by Pointsè if for every non-closed set A and every èsomeè point x 2 Cl X A n A there is a subset B ç A, such that Cl X B n A = fxg. Simon ësië was rst to study these properties from a General Topological point of view. In particular, he observed that every GO space is WAP; constructed examples of two AP spaces such that their product is not WAP and èunder CHè of two countable AP spaces, such that their product is not AP. Bella ëb1ë, see also ëb2ë discovered strong connections of these properties with higher convergence properties radiality, semiradiality and pseudoradiality. E.g. every semiradial space is WAP, every compact WAP space is pseudoradial, a product of compact WAP and compact semiradial space is WAP. Denition 0.2. A space X is radial èpseudoradialè if for every non-closed set A and every èsomeè point x 2 Cl X A n A there is a transnite sequence S ç A which converges to x. A space X is semiradial if for every non-ç-closed set A there is a transnite sequence S ç A which converges to a point outside A and satises jsj çç. Radial! Semiradial! Pseudoradial. This work was done while the second author was visiting Catania University. He is grateful to Italian colleagues for generous hospitality and to CNR for nancial support.

2 532 A. Bella, I.V. Yaschenko General remarks on AP spaces In ëptë, it was observed that!1 + 1 is not AP. The rst author showed in ëb1ë that every scattered compact AP space is Frçechet-Urysohn. Here we go further by the following Theorem 1.1. Every compact AP space is Frçechet-Urysohn. Proof: If X has not countable tightness then X contains an uncountable free sequence ç èëar3ëè. Since our space is compact, ç has at least one complete accumulation point èindeed, outside çè, but on the other hand no subset of ç can have that point as a unique cluster point. So the tightness of X is countable and X, being a compact AP space, is Frçechet-Urysohn. Theorem 1.2. Every pseudoradial AP space is radial. Proof: Let A ç X and x 2 Cl X A n A. Since X is AP there is a subset B ç A, so that x is the only point in Cl X B n A. Since X is pseudoradial, there is a transnite sequence in Cl X B nfxg converging to x. Clearly, this sequence lies in A and we are done. Recall that a space is submaximal if every dense subset is open or, equivalently, every subset with empty interior is closed and discrete èsee ëacë for a recent survey on submaximal spacesè. Proposition 1.3. Every submaximal space is AP. Proof: Let x 2 Cl X A n A. A n Int A has empty interior in X and hence it is closed and discrete in X. Therefore, x 2 Cl X Int A. Since Cl X Int A n A is closed and discrete, there is an open neighborhood W of x, so that Cl X W ë ècl X Int A n Aè =fxg: It is easy to see, that B = W ë Int A satises Cl X B n A = fxg. Countable AP and WAP spaces In ësië Simon used CH to construct two countable AP spaces whose product is not AP, leaving it open, whether the same can be done without CH. We present here twozfc examples ètheorem 2.1 and Example 2.5è, with some stronger properties the rst is the product of a convergent sequence with an AP space which is even non-wap, and the second is a non-ap product of a Frçechet-Urysohn space and a compact metric space. Theorem 2.1. There is a countable AP space C such that the product of C with a convergent sequence is not WAP. From Theorem 2.1 we can get the following

3 On AP and WAP spaces 533 Corollary 2.2. There is a countable non WAP space. Theorem 2.1 is an immediate corollary of Proposition 1.3 and Theorem 2.3. In fact, one may take for C an arbitrary countable submaximal space without isolated points. At least one such a space exists due to van Douwen ëvdë. Theorem 2.3. The product of a countable submaximal dense-in-itself space with a convergent sequence is not WAP. Proof: Let X be the product of a countable submaximal space C with the usual convergent sequence! +1. Fix an arbitrary bijection f : C!!. The set A = fèx; fèxèè : x 2 Cg is not closed because Cl X A = A ë C f!g. Suppose that there is B 0 ç C so that for B = fèx; fèxèè : x 2 B 0 g we have j Cl X B n Aj = 1. Evidently, Cl X B n A = fèy;!èg, for some y 2 C. Since jb ëfyg!j ç1, it follows that y 2 Cl C èb 0 nfygè. So, B 0 has non-empty interior in C. Taking any arbitrary point z 2 Int B 0 nfyg, we see that èz;!è 2 Cl X B n A a contradiction. Observe that the space C above is far from being Frçechet-Urysohn or sequential. Indeed, there is no chance to have a sequential example, because of the following Proposition 2.4. The product of a sequential space X with a countably compact WAP space Y is WAP. Proof: Let A ç X Y be a non-closed subset. Take èx; yè 2 Cl XY A n A. If A ëfxgy were non-closed, we could immediately apply the WAP property of Y to nd a set B ç A satisfying Cl XY B n A = èx; yè. So, consider the case that A ëfxg Y is closed. Passing to a suitable subset, we can assume that A ëfxg Y = ;. Since x 2 Cl X ç X èaè, it follows that ç X èaè is not closed in X. So, there is a convergent sequence fx n : n 2!g ç ç X èaè with limit outside ç X èaè. For each n 2! take a point y n 2 Y in such a way that èx n ;y n è 2 A. Since countably compact WAP spaces are sequentially compact èëb1ëè, fy n : n 2!g contains a convergent subsequence fy nk : k 2!g. Clearly, fèx nk ;y nk è:k 2!g converges to a point outside A and we are done. We now show that property APmay disappear even in a product of a Frçechet- Urysohn space with a compact metric space. Example 2.5. There are a countable Frçechet-Urysohn space and a countable compact metric space, whose product is not AP. Proof: Let S! = fg ë fèn; mè :n; m 2!g be the familiar countable Frçechet- Urysohn fan having as the unique non-isolated point èhere n stands for the number of a sequence converging to and m for the number of a point in the corresponding sequenceè. We claim that the product X = S! è! +1è 2 of the Frçechet-Urysohn space S! with the compact metric space è! +1è 2 is not AP. Indeed, let A = fèèn; mè; èn; mèè : n; m 2!g be a "diagonal" over sets of isolated points in both factors. Clearly, è; è!;!èè 2 Cl X A. If the product were AP then there should exist B ç A, with Cl X BnA = fè; è!;!èèg. Since, 2Cl S! ç S! èbè,

4 534 A. Bella, I.V. Yaschenko there exists n 2! so that ç S! èbè intersects the n-th sequence in an innite set. Now, it is easy to realize that è; èn;!èè 2 Cl X B n A and therefore X cannot be an AP space. AP and WAP properties in function spaces We start with a bit technical result. We shall use the following Denition 3.1 èëar2ëè. A space X has countable fan-tightness provided that for every point x 2 X and every countable family G n of subsets of X if x 2 Cl X G n for any n 2! then there are nite F n ç G n, so that x 2 Cl X ëff n : n 2!g. Denition 3.2. A space X is!-è-monolithic if the closure of every countable subset has countable pseudocharacter èin itselfè. Theorem 3.3. Every!-è-monolithic space with countable fan-tightness is AP. Proof: Let x 2 Cl X A n A. Since X has countable tightness, we can assume that A is countable. Since X is!-è-monolithic, èèx; Cl X Aè ç!. So, there is a countable decreasing family W n : n 2! of neighborhoods of x in Cl X A, so that ëfcl X W n : n 2!g = fxg. Evidently, x 2 Cl X èw n ë Aè. Since X has countable fan-tightness, there are nite F n ç W n ë A, such that x 2 Cl X èëff n : n 2!gè. Let F = ëff n : n 2!g. Since for each n 2! F n W n is nite, F cannot have cluster points other than x. So, j Cl X F n Aj = fxg. From Theorem 3.3 we can immediately get the following two propositions. Corollary 3.4. Every space with countable pseudocharacter and countable fantightness is AP. Corollary 3.5. Every!-monolithic space with countable fan-tightness is AP. Awell known result concerning function spaces with the topology of pointwise convergence says that C p èxè isfrçechet-urysohn if and only if it is sequential if and only if it is a k-space èëgnëè. This result was further improved in ëgnsë, where it was shown that C p èxè is radial if and only if it is Frçechet-Urysohn. A natural question is now whether property AP could be added to the previous list of equivalences. It turns out that this is not the case, even if we assume that C p èxè is countably AP èi.e. countably tight and APè. A simple example witnessing this is C p èiè, where I is the unit segment. More generally, wehave the following Theorem 3.6. If X is a ç-compact space then C p èxè is countably AP. Proof: Every ç-compact space is a Hurewicz space. Hence C p èxè has countable fan-tightness èëar2ëè. Also, a ç-compact space is!-stable and hence C p èxè is!-monolithic èëar1ëè. Now apply Corollary 3.5. Notice that the last result cannot be strengthened to Lindelíof--spaces.

5 On AP and WAP spaces 535 Example 3.7. C p over a separable metric space can fail to be WAP. Proof: By Corollary 2.2 there is a countable non WAP space X. Then Y = C p èxè is separable and metric. Furthermore, X can be embedded into C p èy èas a closed subspace by the evaluation mapping. Therefore, C p èy è is not WAP. The next natural question concerns whether AP and WAP coincide in function spaces. Theorem 3.8. Let k é!be a regular!-inaccessible cardinal, i.e. ç! éçfor any çéç. Then C p èkè is WAP but not AP. Proof: It is known èëgnsëè that C p èkè is pseudoradial, but not radial. So, C p èkè is not AP by Theorem 1.2. To reach our target, we improve the above mentioned result of ëgnsë by showing that such ac p èçè is actually semiradial, and hence WAP because of a result in ëb1ë. In ëdtë it was observed that every convergent transnite sequence in C p èçè has length either ç or!. Thus, to check that C p èçè is semiradial it suces to look at the non-ç-closed subsets for ç é ç. Clearly, our result will be achieved if we prove that the closure of every set B ç C p èçè of cardinality less than ç has still cardinality less than ç. Let B 0 be the sequential closure of B. It is well known that we have jb 0 jçjbj! and consequently, asç is!-inaccessible, we have also jb 0 j éç. We claim that B 0 gives the full closure of B in C p èçè. In fact on the contrary, since C p èçè is pseudoradial, there would exist a transnite sequence in B 0 converging to some point outside B 0. But, thanks to the observation in ëdtë, such a transnite sequence must be countable and so it does not converge to any point outside B 0. Notice that the argument in the proof of the above theorem shows that C p èçè is indeed R-monolithic. A space X is R-monolithic if for every non-closed set A ç Cl X B ç X there is a transnite sequence S ç A converging to some point outside A and satisfying jsj çjbj. Thus, Theorem 3.8 answers Question 7 in ëb2ë. Questions These are two key questions in the study of compact WAP spaces: Question 4.1 èëb2ëè. Does there exist a compact pseudoradial non-wap space? Question 4.2 èëb2ëè. Is the product of two ècountably manyè compact WAP spaces still WAP? In view of our study of AP and WAP properties in function spaces the following questions could be interesting: Question 4.3. When is C p èxè AP or WAP? Question 4.4. Suppose C p èxè is AP. Is then C p èxè a space with countable tightness?

6 536 A. Bella, I.V. Yaschenko References ëacë Arhangel'skij A.V., Collins P.J., On submaximal spaces, Topology Appl. 64 è1995è, 219í241. ëar1ë Arhangel'skij A.V., Factorization theorems and function spaces: stability and monolithicity, Soviet Math. Dokl. 26 è1982è, 177í181. ëar2ë Arhangel'skij A.V., Hurewicz spaces, analytic sets and fan-tightness of spaces of functions, Soviet Math. Dokl. 33 è1986è, 396í399. ëar3ë Arhangel'skij A.V., On bicompacta hereditarily satisfying Suslin's condition. Tightness and free sequences, Soviet Math. Dokl. 12 è1971è, 1253í1257. ëb1ë Bella A., On spaces with the property of weak approximation by points, Comment. Math. Univ. Carolinae 35 è1994è, 357í360. ëb2ë Bella A., Few remarks and questions on pseudoradial and related spaces, Topology Appl. 70 è1996è, 113í123. ëvdë van Douwen E.K., Applications of maximal topologies, Topology Appl. 51 è1993è, 125í 140. ëdtë Dimov G., Tironi G., Some remarks on almost radiality in function spaces, Acta Univ. Carolinae 28 è1987è, 49í58. ëgnë Gerlits J., Nagy Z.S., Some properties of CèXè I, Topology Appl. 14 è1982è, 151í161. ëgnsë Gerlits J., Nagy Z., Szentmiklossy Z., Some convergent properties in function space, General Topology and its Relations to Modern Analysis and Algebra VI èz. Frolçk, ed.è, Heldermann Verlag, Berlin, 1988, pp. 211í222. ëptë Pultr A., Tozzi A., Equationally closed subframes and representation of quotient spaces, Cahiers de Topologie et Geometrie Dierentielle Categoriques 34 è1993è, 167í183. ësië Simon P., On accumulation points, Cahiers de Topologie et Geometrie Dierentielle Categoriques 35 è1994è, 321í327. Dipartimento di Matematica, Cittça Universitaria, Viale A. Doria 6, Catania, Italy Moscow Center for Continuous Mathematical Education, B. Vlas'evskij 11, Moscow, Russia èreceived September 8, 1997è