Download Linearly Ordered and Generalized Ordered Spaces

“Linearly Ordered and Generalized Ordered Spaces”
May 6, 2002
For any linearly ordered set (X, <), let I(<) be the topology on X that has
the collection of all open intervals of (X, <) as a base. The topology I(<) is
the open interval topology of the order < and (X, <, I(<)) is a linearly
ordered topological space or LOTS. For a subset Y ⊆ X, it can happen
that the relative topology I(<)|Y on Y does not coincide with the open interval
topology I(< |Y ) induced on Y by the restricted ordering. In some cases, there
might be some other ordering of Y whose open interval topology coincides with
I(<)|Y , but in other cases there is not. Perhaps the best known example is the
Sorgenfrey line, which is homeomorphic to the subspace Y = {(x, 1) : x ∈ R}
of the lexicographically ordered LOTS X = R × {0, 1}. There is no re-ordering
of Y whose open-interval topology coincides with the subspace topology that Y
inherits from X.
The study of subspaces of LOTS in their own right was pioneered by E.
Čech who introduced generalized ordered spaces or GO-spaces, i.e., triples
(X, <, T ) where < is a linear ordering of the set X and T is a Hausdorff topology on X such that each point of X has a T -neighborhood base consisting
of (possibly degenerate) intervals. (Such spaces have also been called suborderable spaces.) The class of GO-spaces is known to coincide with the class
of topological subspaces of LOTS, and for any GO-space (X, <, T ), there is a
canonical linearly ordered set (X ∗ , <∗ ) whose ordering extends the given ordering of X, and that has the property that (X, T ) embeds as a closed subspace of
(X ∗ , I(<∗ )). In addition, the extension space X ∗ “inherits” many, but not all,
of the topological properties of X. For example, X is metrizable if and only if
(X ∗ , I(<∗ )) is metrizable, but there are examples in which the GO-space X is
perfect (= closed subsets of X are Gδ -sets) while X ∗ is not.
Historically, LOTS and GO-spaces have been valuable sources of counterexamples in topology. The usual space [0, ω1 ) of all countable ordinals, the Sorgenfrey line, and the Michael line (i.e., the usual space of real numbers with
each irrational isolated) are perhaps the best known elementary examples. More
elaborate examples can be constructed if one is willing to start with more complicated linear orders, e.g., those derived from lexicographic products [6], or
various tree-to-line constructions [14], including Aronszajn trees and Souslin
trees. In particular, the study of Souslin trees and Souslin lines (i.e., nonseparable LOTS or GO-spaces that have countable cellularity), whose existence
or non-existence is undecidable in ZFC, has been one of the central forces in the
study of ordered spaces during the last century [4].
As can be seen in the survey paper [11], the problems of recognizing which
topological spaces have the open interval topology of some linear order (the orderability problem) and which topological spaces are GO-spaces with respect
to some linear order (the sub-orderability problem) have been studied since
the early years of topology. Topological characterizations of the unit interval
(e.g., as the unique compact metric space that is connected, locally connected,
and has two non-cut points) can be interpreted as orderability theorems, al1
though it is unlikely that their authors would have seen them as such. In
1941, Eilenberg published one of the first modern orderability theorems, proving that a connected, locally connected space X is orderable if and only if
X 2 − {(x, x) : x ∈ X} is not connected. Kowalsky proved another orderability
theorem for connected spaces: a connected, locally connected space X is orderable if and only if given any three connected proper subsets of X, two of them
do not cover X. An orderability theorem for compact connected spaces, given
in Theorem 2-24 of [7] asserts that a compact connected space is orderable if
and only if it has exactly two non-cut-points. Another orderability theorem
for compact connected spaces, based on Michael’s selection theory, is discussed
below.
At the other end of the spectrum from connected spaces are zero-dimensional
spaces. Topological characterizations of spaces of ordinals can be viewed as
orderability theorems. For metrizable zero-dimensional spaces, Herrlich proved
the fundamental orderability theorem in 1965: any strongly zero-dimensional
metric space is linearly orderable. (Lynn had previously proved that result for
zero-dimensional separable metric spaces.) A nice proof of Herrlich’s theorem
appears in [E, Problem 6.4.2]. One interesting corollary of Herrlich’s theorem is
that a certain strange metrizable space described by A.H. Stone [13] is orderable.
Stone’s metric space is uncountable, strongly zero-dimensional, not a union of
countably many relatively discrete subspaces, and has the property that every
separable subspace is countable. Knowing that Stone’s space is orderable has
been the key to several recent examples. Later, in 1977, Purisch gave necessary
and sufficient conditions for the orderability of any metric space. His conditions
are very technical and are related to the earlier work of Herrlich and to work
by Mary Ellen Rudin on orderability of subspaces of the real line.
Special orderability theorems for compact Hausdorff spaces are known. One,
due to deGroot and Schnare in 1972, introduced the notion of nests (i.e., collections of sets that are linearly ordered by inclusion): a compact Hausdorff space
X is orderable if and only if there is a collection of open sets that T1 -separates
points of X and that is the union of two nests. Nests played a central role in the
solution of the general orderability problem by van Dalen and Wattel in 1973.
They first proved that a T1 -space X is sub-orderable (i.e., is a GO-space with
respect to some linear ordering) if and only if X has a subbase that consists of
two nests. They went further, proving that a T1 -space X is orderable if and only
X has a subbase S1 ∪ S2 where
T each Si is a nest with the additional property
that if
T
∈
S
satisfies
T
=
{S ∈ Si : T ⊂ S and T 6= S}, then T also satisfies
i
S
T = {S ∈ Si : S ⊆ T and S 6= T }. Related characterizations were obtained
by Deak and Certanov.
Another type of orderability theorem is based on selection theory. Let 2X
be the space of all non-empty closed subsets of X with the Vietoris topology,
and let X(2) be the subspace {T ∈ 2X : |T | = 2}. In 1951, E. Michael gave
an orderability theorem for compact Hausdorff, connected spaces based on the
existence of continuous selections, and in 1981 van Mill and Wattel improved
Michael’s theorem by deleting the hypothesis of connectedness. They proved
that the following properties of a compact Hausdorff space X are equivalent: X
2
is orderable; there is a continuous function f : 2X → X with f (T ) ∈ T for each
T ∈ 2X ; there is a continuous function f : X(2) → X with f (T ) ∈ T for each
T ∈ X(2). In a subsequent paper, van Mill and Wattel gave a characterization
GO-spaces based on selection theory. For more information and references about
orderability, see [11] or [8].
In ZFC, all GO-spaces have certain very strong topological properties, e.g.,
particularly strong normality properties. For example, any GO-space is monotonically normal and consequently hereditarily collectionwise normal . Another
normality-related property of any GO-space is a version of the Dugundji Extension Theorem. For any space X, let C ∗ (X) denote the vector space of all
continuous bounded real-valued functions on X, equipped with the sup-norm
topology. Heath and Lutzer proved that for any closed subspace A of the GOspace X, there is a linear function e : C ∗ (A) → C ∗ (X) with the property that
e(f ) extends f and the range of e(f ) is contained in the closed convex hull of
the range of f for each f ∈ C ∗ (A). Thus, e is a norm-1 linear extender. The
best recent results on the Dugundji Extension Theorem in GO-spaces are due
to Gruenhage, Hattori, and Ohta; see [3], [8]. (The above normality results hold
in ZFC. By way of contrast, van Douwen has shown that without the Axiom
of Choice, there is a LOTS that is not normal and does not satisfy the the
Dugundji Extension Theorem [5].)
In addition to its normality properties, every GO-space has certain strong
covering properties. For example, each GO-space is countably paracompact.
Furthermore, every GO-space is ω0 -fully-normal, i.e., for any open cover U of
a GO-space X, there is an open refinement
T V with the property that for any
countable
subcollection
V
⊆
V
having
V0 6= ∅, some member of U contains
0
S
V0 . Another covering property of any GO-space X is orthocompactness,
T
i.e. every open cover U has an open refinement V with the property that V1
is open for every subcollection V1 ⊆ V. A final example of a covering property
shared by all GO-spaces is that each open cover U of a GO-space has an open
refinement W = {W (U ) : U ∈ U } satisfying W (U ) ⊆ cl(W (U )) ⊆ U for each
U ∈ U. Many of these covering properties can be proved using the “method
of coherent collections” explained in [8]. Coherent collections also provide a
unified approach to the study of covering properties such as paracompactness
in GO-spaces.
The theory of cardinal functions is well-developed for linearly ordered spaces.
Souslin’s original question, whether a LOTS with countable cellularity must be
separable, has been studied at length, and is now known to be undecidable
in ZFC. In ZFC we now know that if X is a LOTS, then c(X) = hL(X) ≤
c(X × X) = d(X) = hd(X) ≤ c(X)+ , and |X| ≤ 2c(X) . Most of these results
were originally proved by Kurepa and have been rediscovered many times by
others. A unified presentation appears in Todorčević’s paper “Cardinal functions in linearly ordered topological spaces” in [3]. In addition, K.P. Hart showed
that w(X) = c(X) × ψ(X), and that result yields yet another proof that the
Sorgenfrey line cannot be ordered in such a way that it becomes a LOTS.
Paracompactness [E] is an important covering property that some, but not
all, GO-spaces have. The literature contains many weak covering properties
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(e.g., being metacompact, metaLindelöf , θ-refinable, weakly δθ-refinable, and
having the D-space property) that are equivalent to paracompactness in a GOspace [8], [3]. The first characterization of paracompactness in LOTS was given
in the 1950s by Gillman and Henriksen who introduced the notion of a Q-gap
and who proved in essence that a GO-space X is paracompact if and only if
whenever G and H are disjoint open sets that cover X and have the property
that x < y whenever x ∈ G and y ∈ H, then there are closed discrete subsets
C and D of X with the property that C ⊆ G is cofinal in G and D ⊆ H is
coinitial in H. (This is actually Faber’s version [6] of the Gillman-Henriksen
theorem.) Another characterization of paracompactness was given by Engelking
and Lutzer: a GO-space X is paracompact if and only if no closed subspace of
X is homeomorphic to a stationary subset of a regular uncountable cardinal
[8]. (Subsequently, Balogh and Rudin proved that the same result holds for
the more general class of monotonically normal spaces.) A typical use of that
characterization is to show that if P is a closed-hereditary topological property
that no stationary set in a regular uncountable cardinal can have, then P implies
paracompactness in any GO-space. One example of such a P is the property
“X has a point-countable base” and another is “the space X is perfect.”
Metrization theory for LOTS is particularly simple: Lutzer showed that a
LOTS is metrizable if and only if it has a Gδ -diagonal. That theorem does not
hold for GO-spaces, as the examples of the Sorgenfrey and Michael lines show.
(Indeed, the fact that the Sorgenfrey and Michael lines have a Gδ -diagonal and
are not metrizable is an easy proof that there is no re-ordering of the set of real
numbers whose open interval topology coincides with the Sorgenfrey or Michael
line topology.) Metrization theory for GO-spaces is summarized by Faber’s
metrization theorem [6]: a GO-space (X, <, T ) is metrizable if and only if there
is a σ-closed-discrete subset D ⊆ X that is dense in (X, T ) and has the property
that x ∈ D whenever x ∈ X has the property that [x, →) (respectively (←, x])
is in T − I(<).
Over the years, many topological properties have been studied as components of metrizability, and experience shows that many of these properties,
known to be distinct among general topological spaces, are equivalent in the
class of GO-spaces. For example, for any GO-space X, the following are equivalent: (a) X is metrizable; (b) X has a σ-locally countable base; (c) X is
developable; (d) X is semistratifiable. Similarly, for a GO-space the properties
of having a σ-disjoint base or having a σ-point finite base are each equivalent to
being quasi-developable. Finally, Gruenhage has proved that for GO-spaces, the
existence of a point-countable base is equivalent to the Roscoe-Collins property
“open-G.”
Clearly any metrizable space has a σ-disjoint base, and any space with a
σ-disjoint base has a point-countable base. Research has shown that among
GO-spaces, there are interesting topological properties that reverse each of those
general implications. For example, if a GO-space has a σ-disjoint base and is
perfect, or has a σ-disjoint base and is a p-space in the sense of Arhangel’skii,
then the space is metrizable. Finding topological properties that could be added
to the existence of a point-countable base to yield a σ-disjoint base is harder.
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One solution is the following property, called Property III: there is a sequence
hUn i of open subsets of X, and a sequence hDn i where Dn is a relatively closeddiscrete subset of Un , such that whenever G is open and p ∈ G, then for some
n ≥ 1, p ∈ Un and Dn ∩ G 6= ∅. See [3] for more details.
As noted above, the basic metrization theorem for GO-spaces is due to Faber.
However, there are metrization theorems for GO-spaces that do not seem to
follow directly from Faber’s result. For example, Bennett and Lutzer showed
that metrizability of a GO-space X is equivalent to the statement that every
subspace of X is an M-space in the sense of Morita or that every subspace is
a p-space in the sense of Arhangel’skii (see also [15]), and Balogh and Pytkeev
showed that a GO-space must be metrizable if it is hereditarily a Σ-space in the
sense of Nagami. A particularly useful lemma when studying GO-spaces with
Gδ -diagonals is an old result of Przymusinski: If (X, <, T ) is a GO-space with
a Gδ -diagonal, then there is a metrizable topology M ⊆ T having the property
that (X, <, M) is also a GO-space. See [3] for further details and references.
There is a rough parallelism between the metrization theory for compact
Hausdorff spaces and for LOTS. Lutzer’s Gδ -diagonal metrization theorem parallels Šneider’s much earlier theorem for compact Hausdorff spaces (see [E,
4.2.B]), and both theorems generalize to paracompact subspaces that can be
p-embedded in some compact Hausdorff space, or in some LOTS. However, the
parallelism is not complete. For example, Juhasz and Szentmiklossy proved
that, under the Continuum Hypothesis, any compact Hausdorff space with a
small diagonal must be metrizable. A theorem of van Douwen and Lutzer says
that, in ZFC, any Lindelöf LOTS with a small diagonal must be metrizable, but
an example constructed from Stone’s metric space (above) shows that a LOTS
may be paracompact, perfect, Čech-complete, have a small diagonal, and have
a σ-closed discrete dense set without being metrizable. See [3] for details.
A great deal of research has been devoted to studying two closely related
properties of a GO-space, namely having a σ-closed discrete dense set, and being
perfect. It is easy to see that the first implies the second, and that a Souslin
line is a consistent example of a LOTS with the second, but not the first,
property. Maarten Maurice asked whether there is a ZFC example of a LOTS
that is perfect and yet does not have a σ-closed-discrete dense set. A second
question, posed by Heath, asks whether there is a ZFC example of a perfect
non-metrizable LOTS with a point-countable base. Heath’s question is related
to the question of Maurice because it is known that a GO-space with a σ-closeddiscrete dense set and a point-countable base must be metrizable, and because
(as shown by Bennett and Ponomarev) if there is a Souslin line, then there is
a Souslin line with a point-countable base. A third question, posed by Nyikos,
asks whether there is a ZFC example of a non-metrizable non-Archimedean
space (i.e., a space having a base that is a tree when ordered by inclusion) that
is perfect. That is a LOTS question because Purisch showed that a perfect
non-Archimedean space is a LOTS under some ordering. Once again, if there is
a Souslin line, then there is a Souslin line that is a counterexample to Nyikos’
question. Qiao and Tall have shown that the questions posed by Maurice,
Heath, and Nyikos are equivalent to each other [3]. The crucial lemma proved
5
by Qiao and Tall is that any first-countable LOTS has dense non-Archimedean
subspace. That result can be proved for first-countable GO-spaces and it follows
that every perfect GO-space has a dense σ-closed-discrete subspace if and only
if every perfect non-Archimedean space is metrizable. Other results concerning
which perfect GO-spaces have dense σ-closed-discrete subspaces were obtained
by Bennett, Heath, and Lutzer in [2]. For example, they showed that the
following properties of a perfect GO-space X are equivalent: X has a σ-closeddiscrete T
dense subspace; X has a sequence Gn of open covers such that for each
p ∈ X, {St(p, Gn ) : n ≥ 1} is countable; X = Y ∪ Z where both Y and Z are
GO-spaces with a Gδ -diagonal; there is a continuous s-mapping from X into a
metrizable space.
The questions of Maurice, Heath, and Nyikos are also related to an older
question posed by Lutzer, asking whether every perfect GO-space embeds topologically into some perfect LOTS. W. Shi has shown that if the perfect GO-space
X has a σ-closed-discrete subset, then X does embed in some perfect LOTS [3].
As mentioned above, if there is a Souslin line, then there is a perfect LOTS that
does not have a σ-closed-discrete dense set, but we do not know whether there
is a model of ZFC in which each perfect LOTS does have a σ-closed-discrete
dense set. Special cases of Lutzer’s question have been answered in ZFC. Lutzer
pointed out that the Sorgenfrey line is a perfect GO-space that cannot be topologically embedded as a closed subset (or as a Gδ -subset) of any perfect LOTS,
and Shi, Miwa, and Gao gave a perfect GO space that cannot be topologically
embedded as a dense subspace of any perfect LOTS. Other related problems
concerning the possibility of monotone embeddings have been solved by Miwa
and Kemoto but the topological version of Lutzer’s question remains open.
Which topological spaces can be obtained from LOTS or GO-spaces by the
action of various kinds of mappings? Any topological space is the continuous
image of some LOTS, because any topological space is the continuous image
of a discrete space, and any discrete space is a LOTS under some ordering. A
more interesting result in this direction is due to Hušek and Kulpa who proved
that the class of open images of zero dimensional orderable spaces is exactly the
class of “butterfly spaces.” A much harder question was posed by Nikiel who
conjectured that any compact monotonically normal space is the continuous
image of some compact LOTS. Mary Ellen Rudin published a sequence of papers
that culminated in a proof of Nikiel’s conjecture [12].
One particularly important version of the continuous images problem is
called the generalized Hahn-Mazurkiewicz problem and asks which topological spaces are continuous images of arcs. By an arc we mean a compact,
connected LOTS. It is known that the usual unit interval is the unique separable arc, and the original Hahn-Mazurkiewicz theorem characterized continuous
images of the unit interval [0, 1] as being precisely the compact, connected,
locally connected metrizable spaces. A good survey of early work on HahnMazurkiewicz theory is given by Treybig and Ward in [1]. Starting with the
work of Mardešić in the early 1960s, topologists have made steady progress
in understanding which spaces are continuous images of (non-separable) arcs.
Many of the results that they obtained included the hypothesis “Suppose X
6
is the continuous image of some compact LOTS.” Rudin’s theorem, mentioned
above, puts these results in a more natural context. For example, by combining
Rudin’s theorem with a result of Treybig and Nikiel (Theorem 6.6 of [9]), one
would have that a space X is the continuous image of an arc if and only if X
is compact, connected, locally connected, and monotonically normal. Similarly,
Rudin’s theorem combines a theorem of Nikiel and Tymchatyn (Theorem 6.14
of [9]) to say for any homogeneous compact monotonically normal space X, at
least one of the following holds: X is metrizable; X is zero-dimensional; or X
has finitely many components, each of which is a homogeneous simple closed
curve. Other characterizations of continuous images of arcs study the special
structure of certain subcontinua. For example, Nikiel showed that if a connected
compact space X has the property that every connected, compact subspace Y
of X is locally connected, then X is the continuous image of an arc. Ward
and Nikiel characterized continuous images of arcs as being those compact connected spaces that can be approximated, in a certain technical sense, by finite
dendrons, and as being those compact connected spaces that are locally connected and have the property that each of their true cyclic elements can be
approximated by T-sets. See [9] for references and further details. Inverse limit
constructions [10] are often key tools in the study of these spaces.
Rim-properties of a space X (by which we mean topological properties of the
boundaries of members of some open base for X) have also played an important
role in Hahn-Mazurkiewicz theory. For example, Mardešić proved that any
continuous image of an arc is rim-metrizable, and while the converse is not
true, a theorem of Pearson, Tymchatyn, and Ward shows that each rim-finite
connected, compact space is the continuous image of an arc.
Let K be a compact LOTS. The relatively simple structure of such a K
has recently been used to study renorming problems in the Banach space C(K)
with the sup-norm. Haydon, Jayne, Namioka, and Rogers have shown that for
a compact LOTS K, C(K) always has an equivalent Kadec norm, and have
characterized when C(K) has an equivalent locally uniformly convex (LUC)
norm. For example, they show that if L is the lexicographically ordered product
space [0, 1]α , then C(L) has an equivalent LUC norm if and only if α < ω1
and that if M is a compact, connected Souslin line, then C(M ) never has an
equivalent LUC norm. See [3] for more details.
Over the last fifty years, many of the most important examples in product
theory for normal, Lindelöf, and paracompact spaces have been constructed by
fine-tuning certain subspaces of the Sorgenfrey and Michael lines. More recently,
researchers have investigated the product theory of various GO-spaces. Alster
obtained interesting results about products of GO-spaces with Gδ -diagonals.
Kemoto and Yajima extended earier results of B. Scott by showing that if A, B
are subspaces of some space [0, γ) of ordinal numbers, then A×B is normal if and
only if it is orthocompact. Kemoto, Nogura, Smith, and Tamano, and Fleissner
and Stanley obtained definitive results about the equivalence of various covering
properties in a finite product X of subspaces of ordinal spaces, and related
them to the property that X contains no closed subspace homeomorphic to a
stationary subset of a regular uncountable cardinal. See [3] for further details.
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References
[1]
Topology and Order Structures I and II , ed. by H. Bennett and D. Lutzer,
Mathematical Center Tracts 142 and 169, Math Centrum, Amsterdam,
1981 and 1983.
[2]
Bennett, H., Heath, R., and Lutzer, D., Generalized ordered spaces with
σ-closed discrete dense subsets, Proc. Amer. Math. Soc., 129 (2000), 931939.
[3]
Bennett, H. and Lutzer, D., Recent developments in the topology of ordered spaces, in Recent Progress in General Topology ed. by M. Hušek and
J. van Mill, to appear.
[4]
Devlin, K. and Johnsbraten, H., The Souslin Problem, Lecture Notes in
Mathematics no. 405, Springer-Verlag, Berlin, 1974.
[5]
van Douwen, E., Horrors of topology without AC: a nonnormal orderable
space, Proc. Amer. Math. Soc. 95(1985), 101-105.
[6]
Faber, M.J., Metrizability in generalized ordered spaces, Math Centre
Tracts, no. 53, Amsterdam, 1974.
[7]
Hocking, J. and Young, G., Topology, Addison-Wesley, Reading, MA,
1961.
[8]
Lutzer, D., Ordered topological spaces, in Surveys in General Topology
ed. by G.M. Reed, Academic Press, New York, 1980, pages 247-296.
[9]
Mayer, J. and Oversteegen, L., Continuum Theory, pages 453-492 in
[HvM].
[10] Nikiel, J., Tuncali, H., and Tymchatyn, E., Continuous images of arcs and
inverse limit methods, Memoires Amer. Math. Soc. 104, no. 498, 1993.
[11] Purisch, S., Orderability of topological spaces, Topology Atlas Invited
Contributions, http://at.yorku.ca/z/a/a/b/14.htm
[12] Rudin, M.E., Nikiel’s conjecture, to appear.
[13] Stone, A., On σ-discreteness and Borel isomorphism, Amer. J. Math. 85
(1963), 655-666.
[14] Todorčević, S., Trees and linearly ordered sets, pages 235-294 in [KV].
[15] van Wouwe, J., GO-spaces and generalizations of metrizability, Math Centre Tracts, no 104, Amsterdam, 1979.
Harold Bennett and David Lutzer
Texas Tech University and College of William and Mary
8