Download An internal characterisation of radiality

An internal characterisation of radiality
Robert Leek
DPhil candidate, University of Oxford
[email protected]
www.maths.ox.ac.uk/people/profiles/robert.leek
Presented at the 16th Galway Topology Colloquium
Paper has now been submitted; preprint is available here
Some of my recent work has been focused on trying to understand why certain convergence properties arise,
in the hope of gaining a deeper understanding of these spaces.
y
x
Fréchet-Urysohn space
(x n )n<ω → x
Radial space
(y α )α<γ → y
One of the classic examples of a convergence property is of being Fréchet-Urysohn.
Definition 1 (Fréchet-Urysohn). A topological space X is said to be Fréchet-Urysohn at a point x if whenever x ∈ A,
there exists a sequence contained in A that converges to x. If X is Fréchet-Urysohn at every point, it is said to be a
Fréchet-Urysohn space.
We can generalise this by replacing ‘sequence’ with ‘some transfinite sequence’.
Definition 2 (Radial). A topological space X is said to be radial at a point x if whenever x ∈ A, there exists a transfinite sequence (a net whose domain is well-ordered) in A that converges to x. If X is radial at every point, it is said
to be a radial space.
Let us look at some examples of these spaces:
First countable space
Well-based space
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Definition 3 (First countable). A point x in a topological space X is said to be first countable if it has a countable
neighbourhood base.
As topologies are closed under finite intersections, it is easy to show that first countable spaces have descending
neighbourhood bases. Let us call this property ‘well-based’.
Definition 4 (Well-based). A topological space X is well-based at a point x if it has a neighbourhood base wellordered by ⊇.
Thus every first countable point is both Fréchet-Urysohn and well-based and every Fréchet-Urysohn or wellbased point is radial.
We can think of well-based points as having neighbourhoods ‘generated by one nest (a set linearly-ordered
by inclusion)’. What about points ‘generated by two nests’? Let us consider LOTS (Linearly Ordered Topological
Spaces) or GO (Generalised Ordered) spaces.
LOTS (or GO-space)
The neighbourhoods are generated by two nests. It is easy to show that this point is radial, simply because we
can split the space into two halves by contracting along nests. In these two halves, the point will be well-based and
if it is in the closure of a subset A, it must be in the closure of A with respect to one of those halves. Let’s look at
another example of a space with a point ‘generated by two nests’.
ω+1
(ω, ω1 )
ω1 + 1
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Consider the Tychonoff plank and its corner point, (ω, ω1 ). It also has a neighbourhood ‘generated by two
nests’; one coming from the ω + 1 factor and the other coming from the ω1 + 1 factor. However, it is not radial because for any such transfinite sequence, its projections must also converge. This is not possible to due conflicting
cofinalities in the different factors. So what makes one space radial and the other not?
Before answering this question, let’s formalise what we mean by a neighbourhood being ‘generated by nests’.
[Note that from this point onwards, all spaces are assumed to by T1 , although all results hold with minor modifications to definitions].
Definition 5 (Nest system). Let x be a point in a topological space X and let (C i )i ∈I be a non-empty family of nests
of neighbourhoods of x. We call (C i )i ∈I is a nest system for x if
½
\
C i : ∀i ∈ I ,C i ∈ C i
¾
i ∈I
is a neighbourhood base for x.
Note that we do not take finite intersections, like we would for a neighbourhood base in the Tychonoff product.
By analysing LOTS, we can see that when we split this space up into two halves, we can take neighbourhoods from
each of these and glue them together to form a neighbourhood base. A nest system with this property is called
independent.
Definition 6 (Independence). Let (C i )i ∈I be a nest system for x in a topological space X and define for all i ∈ I , the
i th spoke to be
\ [
S(C , i ) :=
Cj
j ∈I \{i }
We call a nest system independent if for all i ∈ I and C i ∈ C i
\
i ∈I
Ci =
[
(C i ∩ S(C , i ))
i ∈I
C1 ∈ C1
C 1 ∩C 2 ∩C 3
C2 ∈ C2
C3 ∈ C3
Independence
It can easily be shown that every point with an independent nest system is radial via a similar argument as
given for LOTS. Below, we show the various relations between these properties.
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Radial
Fréchet-Urysohn
Independently-based
LOTS
GO space
Well-based
The question begging to be asked is whether radiality and independently-based coincide. Before tackling this
question, let’s look at another example of an independently-based space and see how two nest systems interact.
Consider the plane R2 and define for every θ ∈ [0, 2π)
C θ := {R2 \{t (cos(θ), sin(θ)) : t ≥ a} : a > 0}.
C θ is a nest and we define the topology on R2 where the origin has (C θ )θ∈[0,2π) as a nest system and every other
point is isolated. Above, we can see an example of one of the elements of a nest and a basic neighbourhood for the
origin.
Now instead of forming neighbourhoods from nests, we could form them by gluing well-based spaces together.
Definition 7 (Spoke system). Let x be a point in a topological space X and let (S i )i ∈I be a family of subspaces
containing x and are well-based at x (such subspaces are called spokes of x). We call (S i )i ∈I a spoke system if
½
[
¾
B i : ∀i ∈ I , B i is a S i -neighbourhood of x
i ∈I
is a neighbourhood base at x with respect to X . Furthermore, if for distinct i , j ∈ I , S i ∩ S j = {x}, we call (S i )i ∈I
independent.
The equivalence of these two independent approaches is shown in the following lemma:
Lemma 8. Let x be a point in a topological space X .
1. Let C = (C i )i ∈I be an independent nest system for x. Then (S(C , i ))i ∈I is an independent spoke system for x.
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2. Let (S i )i ∈I be an independent spoke system for x and define for each i ∈ I
½
¾
[
: B is a S i -neighbourhood for x
C i := B ∩
j ∈I \{i }
Then (C i )i ∈I is an independent nest system for x with respect to X .
Returning to our space in question, it is easy to see that the origin has an independent spoke system of size c.
We may ask if there is one of strictly smaller cardinality. To understand how two spoke systems interact, we use the
following lemma:
Lemma 9 (Reflection lemma). Let x be a point in a topological space X and let (S i )i ∈I , (T j ) j ∈I be spoke systems for
x. Let i ∈ I be given and define K i := { j ∈ J : x ∈ (S i ∩ T j )\{x}}. Suppose:
1. (T j ) j ∈J is independent.
2. x is first countable in S i .
Then K i is finite.
Suppose there exists an independent spoke system (Ti )i ∈I for (0, 0) with |I | < c. Without loss of generality,
assume that the origin is not isolated in each of the Ti ’s. By the reflection lemma and cardinal arithmetic, we can
S
find a θ ∈ [0, 2π)\ i ∈I K i . Then for each i ∈ I , (0, 0) ∉ (S θ ∩ Ti )\{(0, 0)}, where S θ is the θth spoke of C = (C θ )θ∈[0,2π) ,
S
so there exists C i a Ti -neighbourhood for (0,0) that misses S θ \{(0, 0)}. Then ( i ∈I C i ) ∩ S θ = {(0, 0)}, showing that
(0, 0) is isolated in S θ – a contradiction.
How do we go about showing a radial point is independently-based? If it is isolated, this is easily done, but
suppose otherwise. By radiality, we could keep finding transfinite sequences that converge to the point but are
pairwise disjoint in their range. By taking a maximal family of disjoint converging transfinite sequences, it is easy
to show that their union, together with the point, would be a neighbourhood for the point. So it would seem the
collection of subspaces given by the range of a transfinite sequence and the point would form an independent nest
system for x.
However, we will also need to check that when we restrict to a choice of tails, their union is still a neighbourhood. If not then we could find a new transfinite sequence converging ‘along the initial segments’ of the point.
One way to view this would be to take the Riemann sphere and quotient out along {0, ∞}. The original sequences
would converge to 0 whilst the new sequence would converge to ∞.
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So it might not be the case that every radial point is independently-based, but we could create spokes that
intersect ‘far away’. This would then give us a characterisation of radiality.
Theorem 10. Let x be a point in a topological space X . Then x is radial if and only if there is a spoke system (S i )i ∈I
for x such that for every distinct pair i , j ∈ I , x ∉ (S i ∩ S j )\{x}.
Sketch proof. A transfinite sequence is said to converge strictly to a point if the point is not in the closure of any
initial segment of that sequence. We note here that radial points have the following property: if they are in the
closure of a set, then we can find an injective, transfinite sequence from that set that converges strictly to the
point.
Define
T := { f a transfinite sequence in X \{x} : | dom( f )| ≤ |X |, f is injective and converges strictly to x},
A := {F ⊆ T : ∀ f , g ∈ F distinct, f −1 [ran( f ) ∩ ran(g )] is bounded in dom( f )}.
By Zorn’s lemma, we can pick a maximal element F ∈ A and define for all f ∈ F , S f := ran( f ) ∪ {x}. Then by
maximality, (S f ) f ∈F has the desired properties.
The other direction is trivial and only requires the existence of a spoke system.
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The diagram on the previous page demonstrates such a nest system for a neighbourhood point and gives an
example of some disallowed behaviour.
Thus radial points are ‘almost independently-based’. However, this is the best we can do. We will now construct
a Fréchet-Urysohn space that is not independently-based. For every point x ∈ R2 \{(0, 0)}, define C x to be the circle
with centre x and passing through the origin. Let d be the Euclidean metric on the plane and define for every
x ∈ R2 \{(0, 0)} and all ε > 0, A x,ε := C x ∩ B d ((0, 0), ε). Topologise the plane by letting
½
[
A x,εx : ∀x ∈ R2 \{(0, 0)}, εx > 0
¾
x∈R2 \{(0,0)}
be a neighbourhood base at the origin and letting every other point be isolated. Then the origin has a spoke system,
(C x )x∈R2 \{(0,0)} . Moreover, it can be shown that the space is Fréchet-Urysohn.
Now suppose that the origin is independently-based. Then using the reflection lemma, we can find for every
non-origin point x a positive real εx such that for distinct nonorigin points x, y, A x,εx ∩ A y,ε y = {(0, 0)}. Using the
Baire category theorem argument, we can show that this leads to a contradiction.
We may be able to characterise the independently-based spaces within the class of radial spaces. Perhaps they
are the radial spaces ‘described by nests’, whatever that phrase may mean.
Some future directions would be to investigate other classes of spaces, both with weaker forms of convergence
properties and using different classes of nets.
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