Topology Atlas Document # iaai-26

Open problems in topology, seventh status report

Elliott Pearl

Originally published as Open problems in topology, seventh status report, Topology and its Applications 114 no. 3 (2001) 333--352. Copyright © 2001 Elsevier Science B.V. Reprinted with permission.

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Introduction

This is the seventh in a series of status reports on the 1100 open problems listed in the volume Open Problems in Topology (North-Holland, 1990), edited by J. van Mill and G. M. Reed [69, 70, 71, 72, 73, 74, 75]. The goal is to give a brief account of problems solved, by whom, and whether preprints or articles are available. 180 solutions and 54 partial or consistent solutions have been announced so far.

These status reports are intended to support the volume as a source book for current open problems. Anyone who solves a problem is requested to notify both the author(s) of the paper in which it appeared in the volume and one of the two editors of the volume.

Solutions

Problem 5. Yes. Solved by J. Baker and K. Kunen. Baker and Kunen proved that if k is a regular cardinal, then there is a weak Pk+-point in U(k), the space of uniform ultrafilters on k. A preprint is available [10]. Problem 5 only asked for the case k = w1. The weak Pk+-point problem is still open for singular cardinals k.

Problem 19. Yes. Answered by G. Gruenhage and J. Moore [52]. There is an w-Toronto space. An a-Toronto space is defined to be a scattered space of Cantor-Bendixon rank a which is homeomorphic to each of its subspaces of rank a. Gruenhage and Moore constructed countable a-Toronto spaces for each a <= w. Gruenhage also constructed consistent examples of countable a-Toronto spaces for each a < w1.

Problem 20. Yes. J. Stepr¯ans constructed a homogeneous, idempotent filter on w.

Problem 57. Yes. Solved by Z. Balogh. By a theorem of M. E. Rudin, the existence of a normal screenable nonparacompact space implies the existence of a normal, screenable space which is not collectionwise normal [92]. Balogh's example [12], together with Rudin's theorem, answers Problem 57. Balogh's example provides a positive answer to Problem 119,

Problem 97. Yes. Answered by K. Smith and P. Szeptycki [101]. Smith and Szeptycki showed that, assuming \diamondsuuit*, paranormal spaces of character <= w1 are w1-collectionwise Hausdorff. A space is defined to be paranormal if every countable discrete collection of closed sets can be expanded to a locally finite collection of open sets. Both countably paracompact spaces and normal spaces are paranormal. Consequently, assuming \diamondsuit*, countably paracompact first countable spaces are w1-collectionwise Hausdorff, answering Problem 97 in the affirmative.

Problem 110. No. Answered by either of two constructions by Z. Balogh. Problem 110 asks if it is consistent that meta-Lindelöf collectionwise normal spaces are paracompact. Balogh's examples are in ZFC. Balogh constucted a hereditarily meta-Lindelöf hereditarily collectionwise normal space which is not countably metacompact. Balogh constructed a meta-Lindelöf, collectionwise normal, countably paracompact space which is not metacompact.

Problem 116. In Problem 116, S. Watson asked for a ZFC example of a Dowker space of cardinality less than \alephw. In Problem 319, M. E. Rudin asked for a Dowker space of cardinality \aleph1. Rudin remarked that "I would be equally happy to see a Dowker space of cardinality c".

Using pcf theory in ZFC, M. Kojman and S. Shelah proved the existence of a Dowker space of cardinality \alephw+1 [62]. Z. Balogh gave a ZFC example of a Dowker space of cardinality c [11], answering Rudin's alternate problem.

Problem 119. Yes. Answered by Z. Balogh [12]. See Problem 57.

Problem 144. Yes. Answered by R. Grunberg, L. Junqueira and F. D. Tall [53]. Any strengthening of the topology on the real line which is locally compact, locally countable, separable and collectionwise normal is an example of a collectionwise normal space which can be made nonnormal by countable chain condition forcing. The Eric (van Douwen) Line is such a strengthening [40].

Problem 145. Yes, consistently. Answered by R. Grunberg, L. Junqueira and F. D. Tall [53]. Suppose there is an uncountable regular k such that k<k = k. Then there is a nonnormal space X and a countably closed cardinal-preserving P such that P forces X to be normal.

Problem 149. No. Answered by R. Grunberg, L. Junqueira and F. D. Tall [53]. Adding a Cohen subset of w1 with countable conditions will destroy the normality of a non-\aleph1-collectionwise Hausdorff space. In particular, this countably closed forcing does not preserve the hereditary normality of Bing's Example G.

Problem 162. No. M. Tkachenko, V. Tkachuk, R. Wilson and I. Yaschenko [104] proved that no T1-complementary topology exists for the maximal topology constructed by E. K. van Douwen on the rational numbers [41].

Problem 172. Yes. Answered by J. Harding and A. Pogel [55]. They proved that every lattice with 1 and 0 can be homomorphically embedded in the lattice of topologies on some set. S. Watson said that this "is the most important question in this section".

Problem 201. Yes. Answered by S. Shelah and J. Stepr¯ans. It is consistent with MA + not CH that a totally non-trivial (= nowhere trivial) automorphism exists. A preprint is available [98].

Problem 223. No, consistently. A. Bella, A. Baszczyk and A. Szymanski [20] proved that if X is compact, extremally disconnected, without isolated points and of p-weight \aleph1 or less then X is an AR for extremally disconnected spaces iff X is the absolute of one of the following three spaces: the Cantor set, the Cantor cube w12, or the sum of these two spaces. This provides a negative answer to Problem 223 under CH.

Problem 229. Partially solved by E. Coplakova and K. P. Hart [30]. Coplakova and Hart proved that if the bounding number b equals c then there exists a point p in Q* (the Cech-Stone remainder of the space of rational numbers) such that p generates an ultrafilter in the set-theoretic sense on Q and such that p has a base consisting of sets that are homeomorphic to Q.

Problem 240. Yes. I. Farah proved a generalization of Problems 240 and 241 [46].

Theorem Assume Z is a bN-space, X is compact, k is an arbitrary cardinal and f : Xk --> Z. Then Xk can be covered by finitely many clopen rectangles such that f depends on at most one coordinate on each one of them.

Problem 241. Yes. Proved by I. Farah [46]. See Problem 240.

Problem 244. S. Shelah and O. Spinas [96] proved that for every n one can have a model in which wn((w*)n) > wn((w*)n+1). This provides some information about Problem 244.

Problem 245. Yes, to the second part of the problem. S. Shelah and O. Spinas showed that wn(w*) > wn(w* x w*) is consistent [97].

Problem 266. A. Dow and K. P. Hart [38] have shown that there are least 14 different subcontinua of bR \ R: 10 in ZFC alone, four more under CH or at least six more under not CH.

Problem 286. No. Answered by T. Eisworth and J. Roitman [45]. CH is not enough to imply the existence of an Ostaszewski space.

Problem 287. Answered by T. Eisworth [44]. Eisworth showed that it is consistent with CH that first countable, countably compact spaces with no uncountable free sequences are compact. Consequently, it is consistent with CH that perfectly normal, countably compact spaces are compact.

Problem 292. It was mentioned in the third status report that M. Rabus proved that it is consistent with MA and t = \aleph2 = c that every \subset*-increasing w1-sequence in P(w) is the bottom part of some tight (w1, w2*)-gap [88].

In the discussion after Axiom 5.6 (p. 151), P. Nyikos wrote: "Of course, the really interesting models are those where b < c, and there Problem 10 (= Problem 292) and its analogue for higher k ( > w1) seem to be completely open".

Z. Spasojevic answered these questions by providing such models [102]. Spasojevic thereby provided new models (where b < c) which contain separable, first countable, countably compact, noncompact Hausdorff spaces. The existence of such spaces is the (still open in ZFC) title problem of Nyikos's article.

Problem 296. Z. Spasojevic [102] showed that p = w1 implies that there is a tight (w1, w1*)-gap in NN, according to Definition 6.8 (p. 157) by P. Nyikos. However, Nyikos misstated the definition of a tight gap for a pair of families A, B in NN. Definition 6.8 should have specified that pair A, B has to be a gap in NN as well. In particular, f < * g for each f in A, g in B. Problem 296, with the corrected definition of a tight gap in NN, is still open.

It is this corrected version, and not the version of Problem 296 stated in the book, that is needed for the construction of a separable, countably compact, noncompact manifold.

Problem 303. No, surprisingly. Solved by D. Fearnley [47]. Fearnley contructed a Moore space with a s-discrete p-base which cannot be densely embedded in any Moore space with the Baire property.

Problem 319. Z. Balogh gave a ZFC example of a Dowker space of cardinality c, answering Rudin's alternate problem. See Problem 116.

Problem 324. Yes. Z. Balogh proved that for every uncountable cardinal k there is a space X such that

  1. the product of X with every metrizable space is normal;
  2. X has an increasing w1-cover with no refinement by fewer than k closed subsets of X.
This implies a postitive answer to Problem 324 and proves the second (and thus all three) of K. Morita's duality conjectures. A preprint is available.

Problem 329. Problem 329 is Michael's Conjecture: There is a Michael space. J. Moore proved that it is consistent that there is a Michael space of weight less than b [78]. Moore also proved that d = cov(Meager) implies that there is a Michael space.

Problem 333. Yes, for part (b) and (d). S. Shelah proved that, consistently, every maximal almost disjoint family has cardinality strictly bigger than the dominating number, that is, a > d. This is one of the oldest problems on cardinal invariants of the continuum. Shelah also proved the consistency of a > u. A preprint is available [95].

Problem 373. Partially solved by W.-X. Shi. Problem 373 asks if every perfect generalized ordered space can be embedded in a perfect linearly ordered space. Shi proved that any perfect generalized ordered space with a s-closed-discrete dense set can be embedded in a perfect linearly ordered space [99].

Problem 374. Y.-Q. Qiao and F. D. Tall [103] have shown that this problem is equivalent to several classic problems of Maurice, Heath and Nyikos and to a corrected version of (the first part) of Problem 175.

Problem 376. No. Solved by W.-X. Shi [100]. Shi constructed an example of a non-metrizable compact linearly ordered topological space, every subspace of which has a s-minimal base. H. Bennett and D. Lutzer had constructed an example that was not compact [21].

Problem 382. No. W. L. Saltsman constructed an example of a complete, connected, countable dense homogeneous (CDH) metric space which is not strongly locally homogeneous (SLH). A preprint is available. Saltsman had constructed, under CH, a connected CDH subset of the plane which is not SLH [94].

Problem 387. Yes, to the first part of the problem. Solved by B. Lawrence. Lawrence proved that all zero-dimensional subsets of R have a homogeneous w-power [65]. A. Dow and E. Pearl proved that all zero-dimensional first countable spaces have a homogeneous w-power [39]. The problem of which zero-dimensional subsets of R have a CDH w-power remains open.

Problem 394. No. Solved by H. P. Chen [28]. Chen constructed a Hausdorff space which is a quotient image of a metric space but which is not a compact-covering quotient image of a metric space. Chen asked whether there exists such a space which is at least regular.

Problem 398. Consistently, the gap between the inductive dimensions for non-separable metrizable spaces can be arbitrarily large. See Problem 399.

Problem 399. Yes, consistently. S. Mrówka constructed an example of a zero-dimensional metrizable space, called nm0, such that under under a particular set-theoretic axiom S(\aleph0), nm0 does not have a zero-dimensional completion [81]. Specifically, under S(\aleph0) each completion of nm0 contains a copy of the interval. In particular, ind nm0 = 0 and, under S(\aleph0), dim nm0 = 1. Mrówka extended this result to show that under S(\aleph0), any completion of (nm0)2 contains copy of the square [82].

J. Kulesza generalized this by showing that under S(\aleph0), every completion of (nm0)n contains an n-cube [63]. In particular, Ind (nm0)n = dim (nm0)n = n under S(\aleph0). This provides answers to Problem 398 and Problem 399.

R. Dougherty proved the relative consistency of the set-theoretic axiom S(\aleph0) [37]. S(\aleph0) has roughly the strength of an Erdös cardinal. Specifically, Dougherty proved that from the Erdös cardinal E(w1+w), S(\aleph0) is consistent and that from S(\aleph0), it is consistent that E(w) exists.

Problem 423. Problem 423, as it appears in the book, was solved by A. N. Dranishnikov and V. V. Uspenskij [42]. R. Pol informed Uspenskij that the problem should have been posed differently.

Problem 423. Let f: X ==> Y be a continuous map of a compactum X onto a compactum Y with dim f-(y) = 0 for all y in Y. Let A be the set of all maps u: X --> I into the unit interval such that u[f-1(y)] is zero-dimensional for all y in Y. Do almost all maps u: X --> I, in the sense of Baire category, belong to A?
H. Torunczyk gave a positive answer under the assumption that Y is countable-dimensional. Uspenskij extended this result to the case when Y has property C [109]. In the general case, the revised problem remains open.

Problem 438. No. S. Ye and Y.-M. Liu constructed a connected metric space with infinite span and zero surjective span. This settles Problem 438 in the negative. The problem remains open for continua.

Problem 445. No. Answered by P. Minc [77]. Minc gave an example of a hereditarily indecomposable tree-like continuum without the fixed point property.

Problem 458. No. J. Prajs announced that the answer is negative. There is an arcwise-connected homogeneous curve that is not locally connected. A preprint is available [87].

Problem 467. Answered by J. Rogers. Rogers proved that if X is a homogeneous, decomposable continuum that is not aposyndetic and has dimension greater than one, then the dimension of its aposyndetic decomposition is one.

Problem 477. W. W. Comfort asked for which cardinals a <= 2c there exists a topological group G such that Gg is countably compact for all cardinals g < a, but Ga is not countably compact.

K. P. Hart and J. van Mill showed a = 2 is such a cardinalunder MAcountable [56]. Under MAcountable, A. Tomita showed that a = 3 is such a cardinal [108] and there are infinitely many such cardinals a < w [106]. Under MAcountable, A. Tomita and S. Watson showed that such cardinals include all a < w, with examples where the witnessing groups contain no nontrivial convergent sequences. A manuscript is being prepared.

Problem 482. Yes. Answered by A. Tomita and S. Watson. They proved that under MAcountable, there are a p-compact group and a q-compact group whose product is not countably compact. A manuscript is being prepared.

Problem 487. No, to part (a) of the problem. V. Malykhin proved that there is a topological group of countable tightness that is not p-sequential for any p in w* [49]. This is a negative answer to both Problem 486 and Problem 487(a).

E. Reznichenko proved that there is a homogeneous space of countable tightness that is not p-sequential for any p in w* [90]. This is a negative answer to Problem 487(a).

Problem 497. No. Answered by P. Gartside, E. A. Reznichenko and O. V. Sipacheva [50]. There is a Lindelöf topological group with cellularity 2\aleph0.

Problem 508. Yes, consistently. M. Tkachenko asked for a ZFC example of a countably compact group topology on the free Abelian group on c many generators. Under CH, Tkachenko had constructed an example that was even hereditarily separable and connected. A. Tomita constructed an example under MAs-centred [107]. A. Tomita and S. Watson constructed an example under MAcountable. A manuscript is being prepared.

Problem 515. No. Solved by K. Kunen and by D. Dikranjan and S. Watson. Kunen proved that there are countably infinite abelian groups whose Bohr topologies are not homeomorphic [64]. Dikranjan and Watson showed that for every cardinal a > 22c there are two groups of cardinality a with nonhomeomorphic Bohr topologies [36]. Both results are in ZFC. These counterexamples answer Problem 515 and Problem 516 in the negative.

Problem 516. No. Solved by K. Kunen and by D. Dikranjan and S. Watson. See Problem 515.

Problem 517. This problem is still open. It was reported previously that Problems 516 and 517 had been answered. See Problem 515.

Problem 523. It was mentioned in the sixth status report that D. Robbie and S. Svetlichnyi found a counterexample to the Wallace problem under CH [91]. A. Tomita produced a counterexample under MAcountable [105].

Problem 535. A special case was solved by Y. -M. Liu and J. -H. Liang [66, 67]. They proved that a continuous L-domain L with a least element is conditionally complete ("bounded complete") iff Is[X --> L] = s[X --> L] for all core compact spaces X.

Problem 540. Solved for some special cases by B. S. Burdick [25]. Problem 540 asks whether iterating the operation of taking the dual topology eventually leads to a mutually dual pair of topologies. Burdick gives an affirmative answer to this problem for several classes of spaces. Some of the special cases covered are: any T1 space (already solved in 1966 by Strecker), the lower Vietoris topology on any hyperspace, the Scott topology for reverse inclusion on any hyperspace, and the upper Vietoris topology on the hyperspace of a regular space. In all these special cases, Tdd = Tdddd, and therefore at most four distinct topologies, T, Td, Tdd, Tddd, can be created by iterating the dual operator.

Problem 549. Problem 549 asks to "find more absorbing sets". There have been three approaches to solving this very general and vague problem. There have been many papers and some of the authors are listed here.

  1. Searching for concrete natural examples of absorbing spaces: J. Baars, T. Banakh, R. Cauty, J. Dijkstra, T. Dobrowolski, H. Gladdines, S. Gul'ko, W. Marciszewski, J. van Mill, J. Mogilski, T. Radul, K. Sakai, T. Yagasaki, M. Zarichnyi.
  2. Constructing absorbing spaces for certain concrete classes: T. Dobrowolski, J. Mogilski, R. Cauty, M. Zarichnyi, T. Radul, J. Dijkstra.
  3. General constructions of absorbing spaces:
    1. The technique of soft maps and inverse systems: M. Zarichnyi, see [18,§2.3].

    2. Producing C-absorbing spaces for [0,1]-stable classes C: T. Banakh, R. Cauty , see [18,§2.4] or [17,§6].

Problem 551. Yes. Solved by M. Zarichnyi [110].

Problem 555. Yes. Answered by A. Chigogidze and M. Zarichnyi. Every n-dimensional C-absorbing set is representable in R2n+1. A proof based on S. Ageev's characterization theorem for Nöbeling manifolds [1] is given in a preprint by Chigogidze and Zarichnyi[29].

Problem 564. No. R. Cauty observed that the open unit ball in l2 enlarged by a subset of the unit sphere that is of a suitable higher Borel complexity yields a (nonclosed) convex set that provides a negative answer to Problem 564 and Problem 565. Cauty's answers were mentioned in the second status report.

A positive answer to Problem 564 is found for a wide class of l-convex sets (including topological groups and closed convex sets in locally convex linear metric separable spaces) and classes C (including almost all absolute Borel and projective classes); see [15], [16,§4.2, §5.3], [18].

Problem 565. No. R. Cauty's example for Problem 564 provides a negative answer to Problem 565.

Problem 567. Yes. Solved by M. Zarichnyi and by T. Banakh and R. Cauty. Problem 567 asks to find an infinite-dimensional absorbing set in R\infty which does not admit a group structure.

Such an absorbing set was constructed by M. Zarichnyi [18,§4.2.D]. This set admits no cancellable continuous operation X x X --> X and thus is not homeomorphic to any convex subset of a linear topological space. A s-compact absorbing space with the same properties was constructed by T. Banakh and R. Cauty [17].

Problem 568. T. Dobrowolski and J. Mogilski note that the notion of l-convexity used in their set of problems is stronger than the usual one. In particular, their l-convex sets are subsets of metric topological groups. Usually, a set is meant to be l-convex if it admits an equiconnecting function. The absolute retract property implies the usual notion of l-convexity.

So, having in mind such a weaker notion, the assumption "l-convex" would be superfluous in Problems 560, 561, 563, 564, 565, and 568. With such a weaker notion of l-convexity, the examples in Problem 567 provide a negative answer to Problem 568. In the sixth status report, it was mentioned that Problem 568 was solved, but that referred to the weaker usual notion of l-convexity. Problem 568 is still open.

Problem 569. No. Solved by T. Banakh. By modifying a counterexample due to W. Marciszewski [68], Banakh constructed a linear absorbing subset of R\infty that is not homeomorphic to any convex subset of a Banach space as well as a linear absorbing subset of l1 that is not homeomorphic to any convex subset of a reflexive Banach space. See [18,§5.5.B]. These provide negative answers to Problem 569 and Problem 570.

Problem 570. No. Solved by T. Banakh. See Problem 569.

Problem 575. Yes. Answered by N. Nhu, J. Sanjurjo and T. An [83]. They proved that Roberts' example is an AR, therefore homeomorphic to the Hilbert cube.

Problem 576. Yes, for a special case. Solved by T. Banakh in the case where W is a subset of a locally convex space and W contains an almost internal point (the latter occurs if W is centrally symmetric) [15].

Problem 588. Problem 588 asks to find interesting (different from S and from that of [35,Ex. 4.4]) s-compact absorbing sets which are not countable-dimensional.

For every countable ordinal a, T. Radul [89] constructed a C-compactum universal for the class of all compacta with dimCX <= a. Using this result, Radul proved that for uncountable many ordinals b there exist non-countable-dimensional pre-Hilbert spaces Db which are absorbing spaces for the class of compacta with dimC less than b. Here, dimC is Borst's transfinite extension of covering dimension which classifies C-compacta.

M. Zarichnyi showed that some absorbing sets for classes of compacta of given cohomological dimension are not countable-dimensional [111].

Problem 603. No. Answered by W. Marciszewski and T. Dobrowolski. Problems 603, 605 and 606 have been answered in the negative. No preprint is available yet.

Problem 604. No. Answered by T. Banakh. Banakh gave an example of a Borel pre-Hilbert space E homeomorphic to E x E but not to Ef\infty. See [14] and [18,§5.5.C].

Problem 605. No. Answered by W. Marciszewski and T. Dobrowolski. See Problem 603.

Problem 606. No. Answered by W. Marciszewski and T. Dobrowolski. See Problem 603.

Problem 608. Yes. Answered by S. Ageev. A preprint is available [1]. The Nöbeling spaces Nk2k+1 are the k-dimensional analogues of Hilbert space. Nk2k+1 is a separable, topologically complete (i.e. Polish) k-dimensional absolute extensor in dimension k (i.e. AE(k)) with the property that any map of an at most k-dimensional Polish space into Nk2k+1 can be arbitrarily closely approximated by a closed embedding (i.e. it is a strong k-universal Polish space). Problem 608 asks if these properties characterize the Nöbeling spaces Nk2k+1, for k > 0. Ageev proved that these properties characterize the the Nöbeling spaces Nk2k+1, for every k > 1. The one-dimensional case was proven by K. Kawamura, M. Levin and E.D. Tymchatyn [60].

Problem 677. Yes. Anwsered by U. H. Karimiov and D. Repovs [59]. Karimov and Repovs proved that:

  1. Every compact metrizable space is weakly homotopy equivalent to a cell-like compactum; and
  2. There exists a noncontractible cell-like compactum whose suspension is contractible.
The second result gives an affirmative answer to Problem 677.

Problem 766. Yes, the conjecture of M. G. Barrat, J. D. S. Jones and M. E. Mahowald is true. Solved by J. Klippenstein and V. Snaith [61].

Problem 810. M. Morimoto's Problems 810, 811, 812, 813 ask if there are smooth one fixed point actions of compact Lie groups (possibly finite groups) on S3, D4, S5, or S8 (respectively). When G is a compact Lie group, if a G-manifold has exactly one G-fixed point then the action is said to be a one fixed point action.

Problem 811. No, for the case of finite groups. Answered by N. P. Buchdahl, S. Kwasik and R. Schultz. See Problem 810.

Problem 812. No, for the case of finite groups. Answered by N. P. Buchdahl, S. Kwasik and R. Schultz. See Problem 810.

Problem 813. Yes. Answered by A. Bak and M. Morimoto. See Problem 810.

Problem 822. Yes. Answered by B. Oliver. Problems 822, 823 and 824 concern the question of which smooth manifolds can occur as the fixed point sets of smooth actions of a compact Lie group on disks (or Euclidean spaces). Complete answers can be given in the case where G is a finite group not of prime power order. Specifically, for a compact smooth manifold F Oliver described necessary and sufficient conditions for F to occur as the fixed point set of a smooth action of G on a disk (or Euclidean space) [84]. Oliver's description of the necessary and sufficient conditions imply affirmative answers to Problems 822, 823 and 824.

In the case where G is of p-power order for a prime p, a compact smooth manifold F occurs as the fixed point set of a smooth action of G on a disk if and only if F is mod p-acyclic and stably complex. This follows from Smith theory and the results of L. Jones [58].

A similar result holds in the case where G is a compact Lie group such that the identity connected component G0 of G is abelian (i.e., G0 is a torus) and G/G0 is a finite p-group for a prime p. Moreover, K. Pawalowski proved that for such a group G, a smooth manifold F without boundary occurs as the fixed point set of a smooth action of G on some Euclidean space if and only if F is mod p-acyclic and stably complex [86].

Problem 823. Yes. Answered by B. Oliver. See Problem 822.

Problem 824. Yes. Answered by B. Oliver. See Problem 822.

Problem 890. No. Solved by R. Cauty. See Problem 981.

Problem 892. No, to the first part. Solved by R. Cauty. See Problem 981.

Problem 894. No. Solved by R. Cauty. See Problem 981.

Problem 899. It was mentioned in the sixth status report that Problem 899 was "solved in the affirmative by S. A. Antonyan". That report referred to the question of whether the Banach-Mazur compacta Q(n) are AR's. Antonyan's result can be found in [5]. This result was also also proven by S. M. Ageev, S. A. Bogatyi and P. Fabel in [4].

Problem 899 further asks whether the Banach-Mazur compacta are Hilbert cubes. Antonyan proved that the Banach-Mazur compactum Q(2) is not a Q manifold [6]. Ageev and Bogatyi proved that the Banach-Mazur compactum Q(2) is not homeomorphic to the Hilbert cube [2, 3].

Problem 900. Yes. Answered by T. Banakh. For part (a), see Theorem 1.3.2 of [18]. This result was reproved afterward by T. Dobrowolski [34]. For part (b), see [13], which also contains the positive answer to the non-separable version of part (a). This implies a positive answer to Problem 554 since each C-absorbing space is an AR with SDAP.

Problem 981. No. R. Cauty constructed a metrizable s-compact linear topological space that is not an absolute retract and such that it can be embedded as a closed linear subspace into an absolute retract [26]. This important result was announced in the fifth status report. This example answers Problem 560, Problem 890, the first part of Problem 892, Problem 894, Problem 981, Problem 982, Problem 984 (b), Problem 985 except for the case of compact spaces, the first part of Problem 988, and Problem 995.

Problem 982. No. Solved by R. Cauty. See Problem 981.

Problem 984. No, to part (b). Solved by R. Cauty. See Problem 981.

Problem 985. No, except for the case of compact spaces. Solved by R. Cauty. See Problem 981.

Problem 986. Yes. Solved by R. Cauty. Every compact convex subset of a linear metric space has the fixed point property. A preprint is available.

Problem 988. No. R. Cauty's counterexample provides a negative answer to the first part of Problem 988 [26]. See Problem 981. W. Marciszewski's counterexample provides a negative answer to the second part [68]. See Problem 996.

Problem 989. No. Answered by W. Marciszewski. See Problem 996.

Problem 993. No. Solved by R. Cauty [27]. J. Grabowski obtained a very elegant and short solution [51].

Problem 995. No. Solved by R. Cauty. See Problem 981.

Problem 996. No. Answered by W. Marciszewski. Marciszewski contructed two counterexamples [68].

Theorem There exists a separable normed space X such that X is not homeomorphic to any convex subset of a Hilbert space. In particular, X is not homeomorphic to a pre-Hilbert space.
Theorem There exists a separable locally convex linear metric space Y such that Y is not homeomorphic to any convex subset of a normed space. In particular, Y is not homeomorphic to a normed space.
The first example gives a negative answer to Problem 988 and Problem 989. The second example gives a negative answer to Problem 996.

Problem 1008. Yes. T. Banakh and K. Sakai answered these and many other questions [93, 19].

Problem 1053. No, consistently and yes, consistently. In his article, A. V. Arkhangel'ski stated that Problem 1053 is equivalent to this question.

Suppose X is a compact Hausdorff space. If there exists a Lindelöf subspace Y of Cp(K) that separates points of X, must X be countably tight?
O. Pavlov proved that, assuming \diamondsuit, there exists a compact Hausdorff space X of uncountable tightness such that Cp(X) contains a separating family which is a Lindelöf space. A preprint is available [85]. Arkhangel'ski had shown that there is a consistent positive answer to Problem 1053 [7].

Problem 1068. Problem 1068 asks if there is a Borel, or even Gd, two-point set in the plane. These questions are still unsolved. Jan J. Dijkstra and Jan van Mill showed that if a two-point set is Gd then it must be nowhere dense in the plane. A preprint is available [32].

Problem 1070. No. Answered by J. Dijkstra and J. van Mill [31] and by D. Mauldin [69]. The problem should have been stated differently.

Problem 1070. Can a compact zero-dimensional partial two-point set always be extended to a two-point set?

The solutions answer the restated problem negatively.

In view of Mauldin's results of [69], it seems more appropriate to ask under what conditions a compact partial two-point set with zero linear Hausdorff measure can be extended to a two-point set. There are some fairly definitive results about this.

In [33], J. Dijkstra, K. Kunen and J. van Mill proved that there exist compact partial two-point sets with linear Hausdorff measure zero (even with Hausdorff dimension zero) that are not contained in a two-point set. Another result is that, assuming MA, if a partial two-point set has the property that the linear Hausdorff measure of its square is zero then it is extendible to a two-point set. Actually, the theorem in [33] is slightly weaker; the result as stated here will appear in a paper by K. Bouhjar and J. Dijkstra [23].

Problem 1084. M. Barge and J. Kennedy asked:

Problem 1084. Let { p1, p2, ..., pn } be a set of n => 2 distinct points in the sphere S2. Is there a homemorphism of S2 \{ p1, p2, ..., pn } such that every orbit of the homeomorphism is dense?

Problem 1085. Is there a homeomorphism of Rn, n => 3, such that every orbit of the homeomorphism is dense?

T. Homma and S. Kinoshita had proven the following theorem in 1953 [57], which provides a negative answer to both Problem 1084 and Problem 1085.

Theorem If X be a locally compact, noncompact, separable metric space then for any continuous self-map of X the set of all points with a dense orbit has empty interior in X.

N. C. Bernardes, Jr. proved a generalization of this theorem to locally compact spaces which are not necessarily metrizable [22]. Bernardes proved that if X is a locally compact Hausdorff space which is not compact and has no isolated points, then for every continuous self map of X, the set of all points with a dense orbit has empty interior in X.

Problem 1085. No. See Problem 1084.

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Table

This report contains a matrix of problem numbers intended to indicate those problems that are still open. On the matrix, a numbered box is shaded in if the problem has been answered absolutely or shown to be independent of ZFC. A numbered box is shaded in lightly if the problem has been answered in part, for a special case, or consistently, since the volume was published.

closed
partial
open

12345678910111213
14151617181920212223242526
27282930313233343536373839
40414243444546474849505152
53545556575859606162636465
66676869707172737475767778
79808182838485868788899091
9293949596979899100101102103104
105106107108109110111112113114115116117
118119120121122123124125126127128129130
131132133134135136137138139140141142143
144145146147148149150151152153154155156
157158159160161162163164165166167168169
170171172173174175176177178179180181182
183184185186187188189190191192193194195
196197198199200201202203204205206207208
209210211212213214215216217218219220221
222223224225226227228229230231232233234
235236237238239240241242243244245246247
248249250251252253254255256257258259260
261262263264265266267268269270271272273
274275276277278279280281282283284285286
287288289290291292293294295296297298299
300301302303304305306307308309310311312
313314315316317318319320321322323324325
326327328329330331332333334335336337338
339340341342343344345346347348349350351
352353354355356357358359360361362363364
365366367368369370371372373374375376377
378379380381382383384385386387388389390
391392393394395396397398399400401402403
404405406407408409410411412413414415416
417418419420421422423424425426427428429
430431432433434435436437438439440441442
443444445446447448449450451452453454455
456457458459460461462463464465466467468
469470471472473474475476477478479480481
482483484485486487488489490491492493494
495496497498499500501502503504505506507
508509510511512513514515516517518519520
521522523524525526527528529530531532533
534535536537538539540541542543544545546
547548549550551552553554555556557558559
560561562563564565566567568569570571572
573574575576577578579580581582583584585
586587588589590591592593594595596597598
599600601602603604605606607608609610611
612613614615616617618619620621622623624
625626627628629630631632633634635636637
638639640641642643644645646647648649650
651652653654655656657658659660661662663
664665666667668669670671672673674675676
677678679680681682683684685686687688689
690691692693694695696697698699700701702
703704705706707708709710711712713714715
716717718719720721722723724725726727728
729730731732733734735736737738739740741
742743744745746747748749750751752753754
755756757758759760761762763764765766767
768769770771772773774775776777778779780
781782783784785786787788789790791792793
794795796797798799800801802803804805806
807808809810811812813814815816817818819
820821822823824825826827828829830831832
833834835836837838839840841842843844845
846847848849850851852853854855856857858
859860861862863864865866867868869870871
872873874875876877878879880881882883884
885886887888889890891892893894895896897
898899900901902903904905906907908909910
911912913914915916917918919920921922923
924925926927928929930931932933934935936
937938939940941942943944945946947948949
950951952953954955956957958959960961962
963964965966967968969970971972973974975
976977978979980981982983984985986987988
98999099199299399499599699799899910001001
1002100310041005100610071008100910101011101210131014
1015101610171018101910201021102210231024102510261027
1028102910301031103210331034103510361037103810391040
1041104210431044104510461047104810491050105110521053
1054105510561057105810591060106110621063106410651066
1067106810691070107110721073107410751076107710781079
1080108110821083108410851086108710881089109010911092
10931094109510961097109810991100

Elliott Pearl
elliott@at.yorku.ca

Date published: June 12, 2001.

Copyright © 2001 Topology Atlas and Elsevier Science B.V.