Rosie Sexton et al - An Ordinal Indexed Hierarchy of Separation Properties.pdf

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An ordinal indexed hierarchy of separation properties
R. A. Sexton and H. Simmons
University of Manchester
rosi.sexton @ btinternet.com
hsimmons @ manchester.ac.uk
Abstract
We rene and stratify the standard separation properties to produce a descend-
ing hierarchy between T 3 and T 1 . The interpolated properties are related to the
patch properties and the Vietoris modications of the parent space.
Key words: separation property, stacking property
200 AMS Classication: 54D10, 54B20
Preamble
We take another look at the standard separation properties
T 3 =⇒T 2 =⇒T 1 =⇒T 0
and show how the step from T 3 to T 2 can be continued below T 2 . The new separation
properties are related to the patch properties of the parent space, and to the nature of its
Vietoris hyperspaces. We attach to each ordinal α three separation properties, α-neat,
α-regular, and α-trim, such that
α-neat =⇒α-regular =⇒α-trim =⇒(α + 1)-neat
S is 0-neat⇐⇒S is empty
S is 0-regular⇐⇒S is T 3
S is 1-neat⇐⇒S is T 2
S is 1-regular⇐⇒S is ??
S is 2-neat⇐⇒S is ??
.
where the later properties seem not to have been described before, but they certainly
become progressively weaker. For instance, the maximal compact topology, Example 99
in [10], is 1-regular but not T 2 . In Section 8 we give a whole family of examples which
illustrate the dierences between the properties we develop. As the survey [7] shows,
many separation properties have been invented. However, in that work none of these
seem to be arranged in an ordinal indexed family. More recently two ordinal hierarchies
have been described in [8]. One of these is related to, but not the same as, our neat
hierarchy. When we were doing the work for this paper we were unaware of [8]. We say
more about the relationship between the two hierarchies at the end of Section 4.
1
for all α. In general, none of these implications is an equivalence. Here we concentrate
on the neat and regular properties, but the trim properties are worth noting. For a T 0
space S we have
We attach to each compact saturated subset Q of a space S a certain operation, a
derivative ∂ Q , on the familyCS of closed sets of S. (At least that is what we do when S
is sober. For a non-sober space we use a more general approach.) This operation can be
iterated through the ordinals, and eventually stabilizes. The length of the iteration gives
an ordinal rank α, and it is this that is used in the hierarchies.
We nd that neatness, being α-neat for some α, decomposes into two properties.
neat = packed + stacked
The rst, being packed, is well known but unnamed, and is concerned with the behaviour
of the patch space of the parent space. The second, being stacked, is concerned with the
behaviour of certain Vietoris hyperspaces of the parent space. It is the stacking properties
that are stratied by the ordinals.
These results are taken from [9], mainly Chapters 8 and 11. That account is writ-
ten from a point-free perspective. However, to make this account accessible to a wider
readership we develop the material in a point-sensitive way. (We do make the occasional
remark about the point-free approach, but these can be ignored if you wish.)
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Derivatives on a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Scott-open lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Neat spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Three interlacing hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Stacking properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7 V-points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8 Boss spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1 Introduction
We outline the aims of this paper, and describe where the results come from. As mentioned
in the preamble, originally we used point-free methods (and, in part, obtained more
general results). Here we use only point-sensitive methods. However, in this introduction
it will be necessary to allude to the point-free approach. If you are not familiar with these
methods, it will not detract from your understanding of the results presented later.
Let S be a topological space. This has familiesOS of open subsets andCS of closed
subsets. Typically we let
U, V, W . . . range overOS
X, Y, Z . . . range overCS
and we write E ◦ for the interior and E − for the closure of an arbitrary subset E⊆S.
We use the standard separation properties T 3 , T 2 , T 1 , T 0 where T 3 = T 0 +regular. As
explained in the preamble, our aim is to show that T 3 and T 2 are the initial steps in a
hierarchy which descends to somewhere above T 1 .
The specialization order on a space S is the comparison of points given by
p≤q⇐⇒p∈q −
2
for p, q∈S. This is always a pre-order, is a partial order precisely when the space is T 0 ,
and is equality precisely when the space is T 1 .
A saturated set is an upper section of this comparison. Every open set is saturated
but there may be many non-open saturated sets. We write E ↑ for the saturation (upwards
closure) of a subset E⊆S.
LetQS be the family of compact saturated sets. We use Q as a typical member of
QS. Observe that the saturation K ↑ of a each compact set K is inQS. Much of what we
do can be seen as an analysis of the wayQS inuences the more general properties of S.
It is a standard exercise that for a T 2 space S we haveQS⊆CS. For later we state
this in the form of a separation property and sketch the proof.
1.1 LEMMA. Let S be a T 2 space and consider p∈S and Q∈QS with p /∈Q. Then
p∈U Q⊆V U∩V =∅
for some U, V∈OS.
Proof. Fix p /∈Q∈QS. For each q∈Q the T 2 separation gives
p∈U q q∈V q U q ∩V q =∅
for some U q , V q ∈OS. Letting q vary through Q produces an open cover of Q which, by
the compactness, renes to a nite cover, and so produces the required U, V∈OS.
This suggests that in a T 2 space S the sets Q∈QS are trying to behave like points.
We remember this when we discuss the V-modications (Vietoris hyperspaces) of a space.
The propertyQS⊆CS of a space doesn’t seem to have a name, so we give it one.
1.2 DEFINITION. A space S is packed if each compact saturated set is closed.
A space S is tightly packed if each compact saturated set is closed and nite.
Another property of T 2 spaces does have a name.
1.3 DEFINITION. For a space S a closed subset Z is irreducible if it is non-empty and
(Z meets U ) and (Z meets V ) =⇒Z meets U∩V
for each U, V∈OS. For each p∈S the closure Z = p − ={p} − is is irreducible, and we
say p is a generic point of Z. A space S is sober if it is T 0 and each of its closed irreducible
sets has a generic point (which, by the T 0 property, is unique).
Almost trivially both the implications
(1) T 2 =⇒T 1 +sober+packed
T 0 +packed =⇒T 1 (2)
hold. Not so trivially both are strict. An appropriate counter-example for (1) can be
found in [10]. (This book doesn’t deal explicitly with sobriety and packedness. The
dissertation [6] lls in some of the missing details.) The two conditions T 1 and sobriety
are incomparable, a space can have one without the other. Notice that a space is T 1 +sober
precisely when each closed irreducible set is a singleton. Such a space need not be T 2 .
3
Both sobriety and packedness can be viewed as desirable properties of a space. Thus
each T 2 space is acceptable, but there are defective spaces. As an attempt to correct one
or other of the defects we can attach to a space S an appropriate space
σ : S−→ s S
π : p S−→S
using a continuous map σ or π. The left hand space s S is the sober reection of S. For
a T 0 space its points are the closed irreducible subsets of S with ‘essentially the same’
topology. The space s S is always sober and, in particular, S is sober precisely when σ
is a homeomorphism. The right hand space p S is the patch space of S. This has the
same points as S but with more open sets. (We simply declare that each Q∈QS is now
closed.) In particular, S is packed precisely when π is a homeomorphism. Unfortunately,
in general, the space p S need not be packed, and the construction has to be repeated.
Both these constructions use point-sensitive methods (that is the standard methods
of point set topology). There are also point-free methods available. We need to say a few
words about these without going into any details.
Let Top be the category of topological spaces and continuous maps. This is connected
with another category Frm of a more algebraic nature. A frame is a certain kind of
complete lattice. These are the objects of Frm and the arrows are the appropriate
morphisms. There is a pair of contravariant functors connecting Top and Frm. For
a space S the topologyOS is a frame. Each frame A has a point space pt(A) in Top
(obtained by a kind of spectral construction) together with a surjective morphism
A−→Opt(A)
which need not be an isomorphism.
Many constructions in Top can be mimicked in Frm. Sometimes this gives essentially
the same results, sometimes it gives better results, and sometimes it just misses the point.
For instance, by converting a space S into its topologyOS and then taking the point space
pt(OS) we obtain the sober reection of S. Thus σ : S−→ s S is one of the units of the
contravariant adjunction between Top and Frm.
There is also a point-free version of the patch construction. For a sober space S
there is a bijective correspondence betweenQS and the Scott-open lters onOS. This
correspondence is discussed in Section 3. For any frame A there is a process of modifying
A by ‘formally adjoining’ its Scott-open lters. This produces a larger frame
A−→P A
and an embedding. The point set content of the construction is explained in Section 2.
We apply this construction to the topology of a space S to obtain a pair of morphisms
f :OS−→POS
g : POS−→O p S
in Frm. The composite is essentially the map
p S−→S viewed as a frame morphism. In
particular, when S is packed, that is
p S = S, we obtain a pair of morphisms
f :OS−→POS
g : POS−→OS
where the composite g◦f is the identity onOS. The other composite f◦g need not be
the identity on POS. An analysis of this led to the result presented here.
4
Roughly speaking we say a space is neat if the embedding f is an isomorphism. The
topological content of this notion is discussed in Section 4. We nd that
(3) T 0 +neat =⇒T 1 +sober+packed
but this is not an equivalence.
What topological properties ensure neatness? From III(1.2)(iii) of [5] we have
T 3 =⇒T 0 +neat
(and, in fact, a more general point-free result). We improved this by weakening the
hypothesis to T 2 , and this with (3) gives (1) above. An analysis of neatness led to the
ordinal indexed hierarchy of separation properties, as described in Section 4 and 5.
The implication (3) is not an equivalence. What more is needed on the right hand
side? The missing property, that of being stacked, is discussed in Section 6. It turns out
that this is concerned with the nature of the V-modications of the parent space.
For a space S the point-sensitive Vietoris hyperspaces use certain collections of subsets
as points. The most common collection isQS, but there are other larger collections. There
is also a point-free version of this construction, originally described in [3, 4]. This produces
an even larger setVS of points which, in general, are not just certain subsets of S. The
stacking property is concerned with the dierences between these various V-modications.
The background to these results is described in more detail in Section 7.
Section 8 contains a collection of examples which illustrate the notions developed here.
2 Derivatives on a space
At rst sight you may think the following gadgets look a little odd.
2.1 DEFINITION. A derivative on a space S is an operator ∂ onCS which is deationary,
monotone, and respects joins, that is
(d) ∂(X)⊆X
(m) Y⊆X =⇒∂(Y )⊆∂(X)
(j) ∂(Y∪X)⊆∂(Y )∪∂(X)
for each X, Y∈CS. Because of (m) the comparison of (j) is, in fact, an equality.
There is one very well known example of a derivative that is worth remembering.
2.2 EXAMPLE. Let S be a T 0 space. Recall that a point p∈X∈CS is isolated in the
closed set if X∩U ={p}for some U∈OS. Let lim(X) be the set of non-isolated points
of X, the limit points of X. This lim is the CB-derivative onCS.
We compare derivatives in a pointwise fashion, that is
∂ 1 ≤∂ 2 ⇐⇒(∀X∈CS)[∂ 1 (X)⊆∂ 2 (X)]
for derivatives ∂ 1 , ∂ 2 (on the same space). Notice also that the composite ∂ 1 ◦∂ 2 of two
derivatives is itself a derivative, and is smaller than its two components.
The CB-derivative is used to extract the perfect part of a closed set. This is done by
iteration, and a similar process is available with any derivative.
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