Open Problems in Topology II

721 Pages·26.134 MB·English

by  Pearl E. (Editor)

Preface
This new book follows the 1990 volume Open Problems in Topology edited
by J. van Mill and G.M. Reed. It builds on the success of original volume by
presenting currently active research topics in topology. The contributions in this
book are entirely new even though some (many?) of its problems may have been
raised in other sources. As with the original volume, the intent is to provide a
source of dissertation problems and to challenge the research community within
and beyond topology. This volume covers a broad range of topics related to
topologyandexaminessometopicsingreaterdepth. Theproblemsinthisvolume
are supposed to reflect the main trends in general topology and its applications
since 1990 and I hope that they will help direct further research.
Thisvolumewaspreparedwiththeinvaluablehelpofseveraleditorialadvisors.
They are Fr´ed´eric Mynard (General Topology), Michael Hruˇs´ak and Justin Tatch
Moore (Set-theoretic Topology), Alejandro Illanes (Continuum Theory), Dikran
Dikranjan(TopologicalAlgebra),JohnC.Mayer(DynamicalSystems),ThomasJ.
Peters (Computer Science), Biagio Ricceri (Functional Analysis), and Vitalij A.
Chatyrko (Dimension Theory). The advisors helped with the selection of contrib-
utors and the review of manuscripts in the respective parts of the volume.
Thisvolumehasaninthpart,Invited Problems,whichisnotorganizedbyany
particular topic. Here I invited several research groups to contribute problems
representative of their interests.
The journal Topology and its Applications will continue to publish regular
reportsonthestatusofproblemsfromtheoriginalbookandfromthisnewvolume.
Please contribute information on solutions to the respective authors and editors.
I want to thank Jan van Mill for his help in getting this project started. I
thank the Fields Insititute in Toronto for their hospitality during the preparation
of this volume.
Elliott Pearl, Toronto, September 2006
v
Open Problems in Topology II – Edited by E. Pearl
© 2007 Elsevier B.V. All rights reserved
Selected ordered space problems
Harold Bennett and David Lutzer
1. Introduction
A generalized ordered space (a GO-space) is a triple (X,τ,<) where (X,<)
is a linearly ordered set and τ is a Hausdorff topology on X that has a base of
order-convex sets. If τ is the usual open interval topology of the order <, then we
saythat(X,τ,<)isalinearly ordered topological space (LOTS).Besidestheusual
real line, probably the most familiar examples of GO-spaces are the Sorgenfrey
line, the Michaelline, the Alexandroff double arrow, andvarious spaces ofordinal
numbers. In this paper, we collect together some of our favorite open problems
in the theory of ordered spaces. For many of the questions, space limitations
restricted us to giving only definitions and references for the question. For more
detail,see[7]. Notablyabsentfromourlistareproblemsaboutorderability,about
products of special ordered spaces, about continuous selections of various kinds,
andaboutCp-theory,andforthatweapologize. Throughoutthepaper,wereserve
thesymbolsR,P,andQfortheusualsetsofreal,irrational,andrationalnumbers
respectively.
2. A few of our favorite things
The most important open question in GO-space theory is Maurice’s problem,
whichQiaoandTallshowed[16]iscloselyrelatedtoseveralotheroldquestionsof
Heath and Nyikos [7]. Maurice asked whether there is a ZFC example of a perfect
GO-space that does not have a σ-closed-discrete dense subset. A recent paper [9]
has shown that a ZFC example, if there is one, cannot have local density ≤ ω ,
1
and what remains is:
Question 1. Let κ>ω1 be a cardinal number. Is it consistent with ZFC that any 1–2?
perfect GO-space with local density ≤ κ must have a σ-closed-discrete dense set?
Equivalently, is it consistent with ZFC that every perfect non-Archimedean space
with local density ≤ κ is metrizable? Is it consistent with ZFC that every perfect
GO-space with local density ≤κ and with a point-countable base is metrizable?
The following question is closely related to Question 1; see [7] for details.
Question 2 (The GO-embedding problem). Let κ > ω1 be a cardinal number. 3?
Is it consistent with ZFC that every perfect GO-space X with local density ≤ κ
embeds in some perfect LOTS? (Note: the embedding map h is not required to be
monotonic and h[X] is not required to be open, or closed, or dense, in the perfect
LOTS.)
LetMbetheclassofallmetrizablespaces. AspaceX iscleavableoverM[1]
ifforeachsubsetA⊆X,thereisacontinuousfA fromX intosomememberofM
suchthatfA(x)(cid:4)=fA(y)foreachx∈Aandy ∈X\A. Thepropertycleavableover
3
4 §1. BennettandLutzer,Selectedorderedspaceproblems
S isanalogouslydefined,whereS istheclassofallseparablemetrizablespaces. It
is known [2] that the following properties of a GO-space X are equivalent: (a) X
is cleavable over M; (b) X has a weaker metrizable topology; (c) X has a Gδ-
diagonal; (d) there is a σ-discrete collection C of cozero subsets of X such that if
x(cid:4)=y are points of X, then some C ∈C has |C∩{x,y}|=1.
For a GO-space X with cellularity ≤ c, each of the following is equivalent
to cleavability of X over S: (a) X has a weaker separable metrizable topology;
(b) X has a countable, point-separating cover by cozero sets; (c) X is divisible
by cozero sets, i.e., for each A ⊆ X, there is a countable collection CA of cozero
subsets of X with the property that given x∈A and y ∈X\A, some C ∈CA has
x∈C ⊆Y \{y}. However, for each κ>c there a LOTS X that is cleavable over
S and has c(X)=κ (Example 4.8 of [2]). Therefore we have:
4? Question 3. Characterize GO-spaces that are cleavable over S without imposing
restrictions on the cardinal functions of X.
5? Question 4. Characterize GO-spaces that are divisible by open sets (in which
the collection C mentioned in (c) above consists of open sets, but not necessarily
cozero-sets).
6? Question 5. Characterize GO-spaces that are cleavable over R, i.e., in which the
cleaving functions fA can always be taken to be mappings into R.
Cleavability over P and Q has already been characterized in [2]. For compact,
connected LOTS, see [11].
7–8? Question 6. Arhangel(cid:2)ski˘ı has asked whether a compact Hausdorff space X that
is cleavable over some LOTS (or GO-space) L must itself be a LOTS. What if X
is zero-dimensional?
Arhangel(cid:2)ski˘ı proved that if a compact Hausdorff space X is cleavable over
R, then X is embeddable in R. Buzyakova [11] showed the same is true if R is
replaced by the lexicographic product space R×{0,1}.
9? Question 7 (Buzyakova). Is it true that a compact Hausdorff space X that is
cleavable over an infinite homogeneous LOTS L must be embeddable in L?
A topological space X is monotonically compact (resp., monotonically Lin-
delo¨f) if for each open cover U it is possible to choose a finite (resp., countable)
openrefinementr(U)suchthatifU andV areanyopencoversofX withU refining
V,thenr(U)refinesr(V). Itisknownthatanycompactmetrizablespaceismono-
tonicallycompact,andthatanysecondcountablespaceismonotonicallyLindel¨of,
and that any separable GO-space is hereditarily monotonically Lindelo¨f [10].
10–13? Question 8. Is every every perfect monotonically Lindelo¨f GO-space separable?
Is every hereditarily monotonically Lindel¨of GO-space separable? If there is a
Souslin line, is there a compact Souslin line that is hereditarily monotonically
Lindel¨of and is there is a Souslin line that is not monotonically Lindel¨of?
Afewofourfavoritethings 5
Question 9. If Y is a subspace of a perfect monotonically Lindelo¨f GO-space X, 14?
must Y be monotonically Lindelo¨f?
Question 10. (a) Is the lexicographic product space X =[0,1]×{0,1} monoton- 15–17?
ically compact? (b) If X is a first-countable compact LOTS, is X monotonically
Lindel¨of? (c) If X is a first-countable monotonically compact LOTS, is X metriz-
able?
Studying properties of a space X off of the diagonal Δ = {(x,x) : x ∈ X}
means studying properties of the space X2\Δ. For example one can show that a
GO-space X is separable if and only c(X2\Δ)=ω.
Question 11 ([6]). Is it true that a GO-space is separable if X2\Δ has a dense 18–19?
Lindel¨of subspace? If X is a Souslin space, can X2 \Δ have a dense Lindel¨of
subspace?
Question 12. In ZFC, is there a non-metrizable, Lindel¨of LOTS X that has a 20?
countable rectangular open cov(cid:2)er of X2\Δ (i.e., a collection {Un×Vn : n < ω}
of basic open sets in X2 with {Un×Vn :n<ω}=X2\Δ)?
Under CH or b=ω the answer to Question 12 is affirmative [5].
1
Question 13. Suppose X is a LOTS that is first-countable and hereditarily para- 21–22?
compact off of the diagonal (i.e., X2 \Δ is hereditarily paracompact). Must X
have a point-countable base? Is it possible that a Souslin space can be hereditarily
paracompact off of the diagonal?
We note that if one considers GO-spaces rather than LOTS in Question 13,
then there is a consistently negative answer. Under CH, Michael [15] constructed
an uncountable dense-in-itself subset X of the Sorgenfrey line S such that X2
is Lindelo¨f. Because S2 is perfect, X2 is perfect and Lindelo¨f, i.e., hereditarily
Lindelo¨f. But X cannot have a point-countable base.
Let PS be the set of irrational numbers topologized as a subspace of the Sor-
genfrey line. It is known [8] that PS is domain-representable, being a Gδ-subset
of the Sorgenfrey line. In fact, the Sorgenfrey line is representable using a Scott
domain [12].
Question 14. Is the Gδ subspace PS of S also Scott-domain-representable? 23?
Question 15 (Suggested by R. Buzyakova). Suppose X is a GO-space that is 24?
countablycompactbutnotcompactandthateverycompactsubsetofX isametriz-
able Gδ-subspace of X. Must X have a base of countable order?
MaryEllenRudin[17]provedthateverycompactmonotonicallynormalspace
is a continuous image of a compact LOTS. Combining her result with results of
Nikiel proves a generalized Hahn–Mazurkiewicz theorem, namely that a topolog-
ical space X is a continuous image of a compact, connected LOTS if and only if
X is compact, connected, locally connected, and monotonically normal.
Question 16. Is there an elementary submodels proof of the generalized Hahn– 25?
Mazurkiewicz theorem above?
6 §1. BennettandLutzer,Selectedorderedspaceproblems
AspaceX isweaklyperfect ifforeachclosedsubsetC ⊆X thereisaGδ-subset
D of X with D ⊆C =clX(D).
26? Question 17 ([3]). Suppose Y is a subspace of a weakly perfect GO-space X.
Must Y be weakly perfect?
27? Question 18 ([4]). Suppose that X is a Lindelo¨f GO-space that has a small
diagonal and that can be p-embedded in some LOTS. Must X be metrizable?
28–29? Question 19 ([13]). Let < be the usual ordering of R. For which subsets X ⊆R
is there a tree T and linear orderings of the nodes of T so that (a) no node of T
contains an order-isomorphic copy of (X,<), and (b) (X,<) is order-isomorphic
to the branch space of T? (Both R and P are representable in this way, but Q is
not.) Which Fσδ-subsets of R are order-isomorphic to the branch space of some
countable tree?
An Aronszajn line is a linearly ordered set that has cardinality ω , contains
1
no order-isomorphic copy of ω , no copy of ω with the reverse order, and no
1 1
order-isomorphiccopyofanuncountablesetofrealnumbers. Suchthingsexistin
ZFC.
30–31? Question 20 ([14]). Can an Aronszajn line be Lindel¨of in its open interval
topology without containing a Souslin line? If an Aronszajn line has countable
cellularity in its open interval topology, must the Aronszajn tree from which the
line comes contain a Souslin tree?
32? Question 21 (Gruenhage and Zenor). Suppose X is a LOTS with a σ-closed-
discrete dense set and a continuous function Ψ: (X2\Δ)×X → R such that if
x (cid:4)= y are points of X, then Ψ(x,y)(x) (cid:4)= Ψ(x,y)(y). (Note that this is weaker
than the existence of a continuous separating family.) Must X be metrizable?
References
[1] A. V. Arhangel(cid:2)ski˘ı, The general concept of cleavability of a topological space, Topology
Appl.44(1992),27–36.
[2] H.Bennett,R.Byerly,andD.Lutzer,Cleavabilityinorderedspaces,Order18(2001),no.1,
1–17.
[3] H. Bennett, R. W. Heath, and D. Lutzer, GO-spaces with σ-closed discrete dense subsets,
Proc.Amer.Math.Soc.129(2001),no.3,931–939.
[4] H.BennettandD.Lutzer,Diagonalconditionsinorderedspaces,Fund.Math.153(1997),
no.2,99–123.
[5] H. Bennett and D. Lutzer, Off-diagonal metrization theorems, Topology Proc. 22 (1997),
Spring,37–58.
[6] H.BennettandD.Lutzer,Continuousseparatingfamiliesinorderedspacesandstrongbase
conditions,TopologyAppl.119(2002),no.3,305–314.
[7] H. Bennett and D. Lutzer, Recent developments in the topology of ordered spaces, Recent
progressingeneraltopology,II,North-Holland,Amsterdam,2002,pp.83–114.
[8] H. Bennett and D. Lutzer, Domain-representable spaces, Fund. Math. 189 (2006), no. 3,
255–268.
[9] H.BennettandD.Lutzer,OnaquestionofMaartenMaurice,TopologyAppl.153(2006),
1631–1638.
References 7
[10] H.Bennett,D.Lutzer,andM.Matveev,The monotone Lindelo¨f property and separability
in ordered spaces,TopologyAppl.151(2005),180–186.
[11] R. Z. Buzyakova, On cleavability of continua over LOTS, Proc. Amer. Math. Soc. 132
(2004),no.7,2171–2181.
[12] K.DukeandD.Lutzer,Scott-domain-representabilityofcertainorderedspaces,Submitted.
[13] W.FunkandD.Lutzer,Branchspacerepresentationsoflines,TopologyAppl.151(2005),
187–214.
[14] W.FunkandD.Lutzer,Lexicographically ordered trees,TopologyAppl.152(2005),no.3,
275–300.
[15] E. Michael, Paracompactness and the Lindelo¨f property in finite and countable Cartesian
products,CompositioMath.23(1971),199–214.
[16] Y.-Q. Qiao and F. D. Tall, Perfectly normal non-metrizable non-Archimedean spaces are
generalized Souslin lines,Proc.Amer.Math.Soc.131(2003),no.12,3929–3936.
[17] M.E.Rudin,Nikiel’s conjecture,TopologyAppl.116(2001),no.3,305–331.
Open Problems in Topology II – Edited by E. Pearl
© 2007 Elsevier B.V. All rights reserved
Problems on star-covering properties
Maddalena Bonanzinga and Mikhail Matveev
Introduction
In this chapter we consider some questions about properties defined in terms
of stars with respect to open covers. If U is a cover of X, and A a subset of
X, then St(A,U) = St1(A,U) = S{U ∈ U : U ∩ A 6= ∅}. For n = 1,2,...,
Stn+1(A,U) = St(Stn(A,U),U). Even if many properties can be characterized
in terms of stars (thus, paracompactness is equivalent to the requirement that
everyopencoverhasanopenstarrefinement([5],Theorem5.1.12),andnormality
is equivalent to the requirement that every finite open cover has an open star
refinement ([5], 5.1.A and 5.1.J)) we concentrate mostly on properties specifically
defined by means of stars.
All spaces are assumed to be Tychonoff unless a weaker axiom of separation
is indicated. If A is an almost disjoint family of infinite subsets of ω, then Ψ(A)
denotes the associated Ψ-space (see for example [4] or [8]). The reader is referred
to [3] for the definitions of small uncountable cardinals such as b, d or p.
Compactness-type properties
A space X is starcompact [6] if for every open cover U there is a finite A⊂X
such that St(A,U)=X. More generally, X is n-starcompact (where n=1,2,...)
if for every open cover U there is a finite A ⊂ X such that Stn(A,U) = X;
X is n1-starcompact if for every open cover U there is a finite O ⊂ U such
that Stn2(SO,U) = X. This terminology is from [16]; in [4], n-starcompact
spaces are called strongly n-starcompact, and n1-starcompact spaces are called
2
n-starcompact. A Hausdorff space is starcompact iff it is countably compact [6];
a Tychonoff space is 21-starcompact iff it is pseudocompact [4]. In general, none
2
of the implications (countably compact) ⇒ 11-starcompact ⇒ 2-starcompact ⇒
2
pseudocompact can be reversed (see [4, 16] for examples), but in special classes
of spaces the situation may be different.
Question 1 ([20], Problem 3.1). Is every 11-starcompact Moore space compact? 33?
2
Under b=c, the answer is affirmative [20].
Question 2 ([25]). Does there exist a ccc pseudocompact space which is not 2- 34?
starcompact?
One can specify the previous question in this way:
Question 3. Does there exist a pseudocompact topological group which is not 35?
2-starcompact?
The first named author acknowledges GNSAGA-CNR, Italy for financial support.
9
10 §2. BonanzingaandMatveev,Problemsonstar-coveringproperties
Indeed, every pseudocompact topological group is ccc.
Lindel¨of-type properties and cardinal functions
AspaceX iscalledstar-Lindel¨of ifforeveryopencoverU thereisacountable
A ⊂ X such St(A,U) = X. More generally, the star-Lindel¨of number of X is
st-L(X) = min{τ : for every open cover U there is A ⊂ X such that |A| ≤ τ and
St(A,U) = X}. It is easily seen that for a T space X, st-L(X) ≤ e(X), this
1
is the reason why the star-Lindel¨of number is also called weak extent [9]. In [4],
star-Lindel¨of spaces are called strongly star-Lindel¨of.
A Tychonoff star-Lindel¨of space can have arbitrarily large extent, but the
extentofanormalstar-Lindel¨ofspacecanequalatmostc,[17];alternativeproofs
were later given by W. Fleissner (unpublished) and in [11]. Consistently, even a
separable normal space may have extent equal to c [12] (obviously, a separable
space is star-Lindel¨of), however these questions remain open:
36–37? Question 4 ([12]).
(1) Does there exist in ZFC a normal star-Lindel¨of space having uncountable
extent? ... having extent equal to c?
(2) Is there, in ZFC or consistently, a normal star-Lindel¨of space having a
closed discrete subspace of cardinality c?
Thenextquestionisanaturalgeneralizationofthediscussionofcloseddiscrete
subspaces.
38–39? Question 5. Which Tychonoff (normal) spaces can be represented as closed sub-
spaces of Tychonoff (normal) star-Lindel¨of spaces?
As noted by Ronnie Levy, every locally compact space is representable as a
closedsubspaceofaTychonoffstar-Lindel¨ofspace. Thisfollowsfromthefactthat
2κ\{point} is star-Lindel¨of for every κ [11].
SaythatX isdiscretelystar-Lindel¨of ifeveryopencoverU thereisacountable,
closed and discrete subspace A⊂X such St(A,U)=X.
40–41? Question6. Howbigcanbetheextentofanormaldiscretelystar-Lindel¨ofspace?
Can it be uncountable within ZFC?
A Ψ-space is a consistent example [17].
Paracompactness-type properties
A space X is called sr-paracompact (G.M. Reed, [20]) if for every open cover
U, the cover {St({x},U) : x ∈ X} has a locally finite open refinement. More
information on sr-paracompactness can be found in [20].
These properties were introduced by V.I. Ponomarev: X is n-sr-paracompact
(wheren=1,2,...)ifforeveryopencoverU,thecover{Stn({x},U)x∈X}hasa
locally finite open refinement; X is (n+1)-sr-paracompact if for every open cover
2
U, the cover {Stn(U,U):U ∈U} has a locally finite open refinement.
Property(a) 11
Question7(V.I.Ponomarev). Forwhichncanoneconstructann-sr-paracompact 42?
Tychonoff space which is not (n−1)-sr-paracompact? (Here n is either 2,3,... of
2
the form m+ 1 where m=1,2,...).
2
Property (a)
A space X has property (a) (or is an (a)-space) if for every open cover U and
for every dense subspace D ⊂X, there is a closed in X and discrete E ⊂D such
that St(E,U)=X [15].
For countably compact spaces, “closed and discrete” means “finite”; a space
X is called acc (which is abbreviation for absolutely countably compact) if for
every open cover U and for every dense subspace D ⊂X, there is a finite E ⊂D
such that St(E,U) = X [13]. Replacing “finite” with “countable” provides the
definition of absolute star-Lindel¨ofness [1].
In many ways property (a) is similar to normality [10, 15, 18, 21] even if
there are exceptions, thus every Tychonoff space can be embedded into a Ty-
chonoff (a)-space as a closed subspace [14] while normality is a closed-hereditary
property. Regular closed subspaces of (a)-spaces are discussed in [22]. A normal
countablycompactspaceneednotbeacc[19]. Everymonotonicallynormalspace
has property (a) [21].
Hereisoneparallelwithnormality: ifX isacountablyparacompact(a)-space,
and Y a compact metrizable space, then X ×Y has property (a) as well [10]. A
space X is called (a)-Dowker if X is has property (a) while X ×(ω +1) does
not [15].
Question 8 ([10]). Do (a)-Dowker spaces exist? 43?
Another parallel with normality is the restriction on cardinalities of closed
discrete subspaces. The classical Jones lemma says that if D is a dense subspace,
and H a closed discrete subspace in a space, then 2|H| ≤2|D|. The situation with
normality replaced by property (a) is discussed in [7, 8, 15, 23, 24]. A separable
(a)-space cannot contain a closed discrete subspace of cardinality c [15].
Question9([7]). Isitconsistentthatthereisan(a)-spaceX havingacloseddis- 44–45?
crete subspace of cardinality κ+ so that κ=d(X) and 2κ <2κ+? In particular, is
it consistent with 2ω <2ω1 that a separable (a)-space may contain an uncountable
closed discrete subspace?
Question 10 ([7]). Is it consistent that there is an (a)-space X having a closed 46–47?
discrete subspace of cardinality κ+ so that κ = c(X)χ(X) and 2κ < 2κ+? In
particular, is it consistent with 2ω < 2ω1 that a first countable ccc (a)-space may
contain an uncountable closed discrete subspace?
An interesting special case of the problem under consideration is property (a)
for Ψ-spaces. It is easily seen that if A is a maximal almost disjoint family, then
Ψ(A) is not an (a)-space. If |A| < p then Ψ(A) is an (a)-space, and there is
a family A of cardinality p such that Ψ(A) is an (a)-space [24]. The question