Protasov.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 5, No. 2, 2004

pp. 191-198

Resolvability of ball structures

I. V. Protasov

Abstract. A ball structure is a triple B = (X, P, B) where X, P
are nonempty sets and, for any x ∈ X, α ∈ P , B(x, α) is a subset of
X, which is called a ball of radius α around x. It is supposed that
x ∈ B(x, α) for any x ∈ X, α ∈ P . A subset Y ⊆ X is called large
if X = B(Y, α) for some α ∈ P where B(Y, α) =

⋃
y∈Y

B(y, α). The
set X is called a support of B, P is called a set of radiuses. Given a
cardinal κ, B is called κ-resolvable if X can be partitioned to κ large
subsets. The cardinal res B = sup {κ : B is κ-resolvable} is called a
resolvability of B. We determine the resolvability of the ball structures
related to metric spaces, groups and filters.

2000 AMS Classification: 54A25, 05A18.

Keywords: ball structures, resolvability, coresolvability.

1. Introduction

Let B1 = (X1,P1,B1) and B2 = (X2,P2,B2) be ball structures, f: X1 −→
X2. We say that f is a ≺-mapping if, for every α ∈ P1, there exists β ∈ P2
such that f(B1(x,α)) ⊆ B2(f(x),β) for every x ∈ X. A bijection f is called
an isomorphism if f and f−1 are ≺-mappings.

The results from [10], [11], [12] show that the ball structures (with the
isomorphisms defined above) are the natural asymptotic counterparts of topo-
logical spaces. A good motivation to study ball structures related to metric
spaces is in the survey [5].

A topological spaces is called κ-resolvable (κ is a cardinal) if it can be parti-
tioned to κ dense subsets. For resolvability of topological spaces and topological
groups see the surveys [3], [4], [9].

Let B = (X,P,B) be a ball structure. A subset Y ⊆ X is called large if
there exists α ∈ P such that X = B(Y,α). The large subsets of ball structure
can be considered as the duplicates of the dense subspaces of topological space.
Given a cardinal κ, we say that B is κ-resolvable if X can be partitioned to κ
large subsets. The resolvability of B is the cardinal



192 I. V. Protasov

resB = sup{κ : B is κ − resolvable}.

A subset Y ⊆ X is called small if X \ B(Y,α) is large for every α ∈ P .
The small subsets of ball structure can be considered as the duplicates of the
nowhere dense subsets of topological space. Assume that every singleton of X
is small. Given a cardinal κ, we say that B is κ-coresolvable if X can be covered
by κ small subsets. The coresolvability of B is the cardinal

coresB = min {κ : B isκ − coresolvable}.

The referee pointed out that the coresolvability can be considered as the
asymptotic duplicate of the Novak number of topological space X

n(X) = min{|U| : U is a cover of X consisting of nowhere dense subsets}.

In this paper we determine (or evaluate) the cardinal invariants res B and
cores B for a wide spectrum of ball structures B related to metric spaces, groups
and filters. We begin with exposition of results (2), continue with proofs (3)
and conclude the paper with comments and open problems (4).

All ball structures under consideration are supposed to be uniform. A ball
structure B = (X,P,B) is called uniform if B is symmetric and multiplicative.
We say that B is symmetric if, for every α ∈ P , there exists β ∈ P such
that B(x,α) ⊆ B∗(x,β) for every x ∈ X and vice versa, where B∗(x,β) =
{y ∈ X : x ∈ B(y,β)}. A ball structure B is called multiplicative if, for any
α,β ∈ P , there exists γ(α,β) ∈ P such that B(B(x,α),β) ⊆ B(x,γ(α,β)) for
every x ∈ X. Note that if B is uniform and Y ⊆ X is large, then there is
γ ∈ P such that B(x,γ)

⋂

Y 6= ∅ for all x ∈ X. For more detailed information
concerning the uniform ball structures as the asymptotic counterparts of the
uniform topological spaces see [11]. Initially, the problem of resolvability of
ball structures was motivated by the following question [1]: can every infinite
group be partitioned onto two large subsets? For the positive answer to this
question see [14] or [15].

2. Results

Let B = (X,P,B) be a ball structure, κ be a cardinal. We say that a subset
Y ⊆ X is κ-crowded if there exists α ∈ P such that |B(y,α)

⋂

Y | ≥ κ for every
y ∈ Y . A ball structure B is called κ-crowded if its support X is κ-crowded.
The crowdedness of B is the cardinal

cr B = sup {κ : B isκ − crowded}.

Define the preodering ≤ on P by the rule α ≤ β if and only if B(x,α) ⊆
B(x,β) for every x ∈ X. A subset P ′ ⊆ P is called cofinal if, for every α ∈ P ,
there exists β ∈ P ′ such that α ≤ β. The cofinality cf B is the minimal
cardinality of the cofinal subsets of P .



Resolvability of ball structures 193

Proposition 2.1. For every ball structure B = (X,P,B), the following state-
ments hold

(i) if B is κ-crowded, then B is κ-resolvable;
(ii) cr B ≤ res B ≤ cr B · cf B;
(iii) if κ is a finite cardinal and B is κ-resolvable, then B is κ-crowded.

By van Douwen-Illanes’ theorem [7], if a topological space is n-resolvable for
every natural number n, then it can be partitioned to countably many dense
subsets. The referee pointed out that the generalization of van Douwen-Illanes’
theorem for the case of countable cofinality lies in [2]. In view of Proposition
2.1 (iii), the following statement can be considered as the analogue of this
generalization.

Proposition 2.2. Let B = (X,P,B) be a ball structure, < κn >n∈ω be an
increasing sequence of cardinals, κ = sup {κn : n ∈ ω}. If B is κn-crowded for
every n ∈ ω, then X can be partitioned in κ large subsets.

Let (X,d) be a metric space. For any x ∈ X, r ∈ R+, put

Bd(x,r) = {y ∈ X : d(x,y) ≤ r}.

The ball structure (X,R+,Bd) induced by the metric space (X,d) is denoted
by B(X,d). A ball structure B is called metrizable if B is isomorphic to B(X,d)
for the appropriate metric space (X,d). By [8], B is metrizable if and only if
B is uniform, connected and cf B ≤ ℵ0. A ball structure B = (X,P,B) is
called connected if, for any x,y ∈ X, there exists α ∈ P such that x ∈ B(y,α),
y ∈ B(x,α).

Theorem 2.3. For every metric space (X,d), res B(X,d) = cr B(X,d) and
X can be partitioned in cr B(X,d) large subsets.

Let G be an infinite group with the identity e and let κ be an infinite cardinal
with κ ≤ |G|. Denote by ℑ(G,κ) the family of all subsets of G of cardinality
< κ containing e. For all g ∈ G, F ∈ ℑ(G,κ), put

Bl(g,F) = Fg, Br(g,F) = gF.

Denote by Bl(G,κ) and Br(G,κ) the ball structures (G,ℑ(G,κ),Bl) and
(G,ℑ(G,κ),Br). Observe that the mapping g 7−→ g

−1 is an isomorphism
between Bl(G,κ) and Br(G,κ). We say that a subset Y ⊆ X is left κ-large
(left κ-small) if Y is large (small) in the ball structure Bl(G,κ). In other words,
Y is called left κ-large if there exists a subset F ⊆ G such that |F | < κ and
G = FY .

Theorem 2.4. Let G be an infinite group, κ be an infinite cardinal and κ ≤ |G|.
Then G can be partitioned in κ left κ-large subsets.

Let B = (X,P,B) be a ball structure. A subset Y ⊆ X is called bounded if
there exist x ∈ X, α ∈ P such that Y ⊆ B(x,α). We say that B is bounded if X
is bounded. Assume that B is unbounded and connected. Then every bounded
subset of X is small. Since X =

⋃

{B(x,α) : α ∈ P} for every x ∈ X, we



194 I. V. Protasov

conclude that cores B ≤ cf B. In particular, cores B(X,d) ≤ ℵ0 for every
unbounded metric space (X,d). On the other hand, the family of all small
subsets of an arbitrary ball structure is an ideal in the Boolean algebra of all
subsets of X (see [11]). Thus, cores B ≥ ℵ0 for every unbounded ball structure
B. Hence, cores B(X,d) = ℵ0 for every unbounded metric space (X,d). The
following theorem shows that cores B could be much more less than cf B.

Theorem 2.5. Let G be an infinite group, κ be an infinite cardinal and κ ≤ |G|.
If κ < cf (|G|), then cores Bl(G,κ) = ℵ0.

Let G be a topological group, C(G) be a family of all compact subsets of
G containing the identity of G. A ball structure (G,C(G),Bl) is denoted by
Bl(G,C). Clearly, Bl(G,C) = Bl(G,ℵ0) for every discrete group G.

Theorem 2.6. Let G be a non-compact locally compact group, then cores
Bl(G,C) = ℵ0.

Let X be a set and let ϕ be a filter on X such that
⋂

ϕ = ∅. For any x ∈ X,
F ∈ ϕ, put

B(x,F) =

{

X\F, if x /∈ F ;
{x}, if x ∈ F ;

and denote by B(X,ϕ) the ball structure (X,ϕ,B).

Theorem 2.7. Let X be a set, ϕ be a filter on X such that
⋂

ϕ = ∅. Then
res B = 1, cores B = min {|ψ| : ψ ⊆ ϕ,

⋂

ψ = ∅}.

3. Proofs

Proof of Proposition 2.1. (i) Choose α ∈ P such that |B(x,α)| ≥ κ for every
x ∈ X. By Zorn lemma, there exists a subset Y ⊆ X such that the family
{B(y,α) : y ∈ Y } is pairwise disjoint and, for every x ∈ X, there exists
y ∈ Y such that B(x,α)

⋂

B(y,α) 6= ∅. Since |B(y,α)| ≥ κ for every y ∈ Y ,
there exists a family ℑ of κ-many pairwise disjoint subsets of X such that
|F

⋂

B(y,α)| = 1 for all y ∈ Y and F ∈ ℑ. In view of uniformity of B and by
choice of Y , every subset F ∈ ℑ is large. Hence, B is κ-resolvable.

(ii) The left inequality follows from (i). Let ℑ be an arbitrary pairwise
disjoint family of large subsets of X. Pick a cofinal subset P ′ ⊆ P with |P ′| =
cf B. For every α ∈ P ′, put

ℑ(α) = {F ∈ ℑ : B∗(F,α) = X},

where B∗(F,α) =
⋃

x∈F
B∗(x,α).

Take any x ∈ X and α ∈ P ′. Since B(x,α)
⋂

F 6= ∅ for every F ∈ ℑ(α), we
have |ℑ(α)| ≤ |B(x,α)|. Hence, |ℑ(α)| ≤ cr B and the right inequality holds.

(iii) Let F1,F2, ...,Fm be pairwise disjoint large subsets of X. Choose α ∈ P
such that B∗(Fi,α) = X for every i ∈ {1,2, ...,m}. Then B(x,α)

⋂

Fi 6= ∅
for all x ∈ X and i ∈ {1,2, ...,m}. It follows that |B(x,α)| ≥ m and B is
m-crowded. ✷



Resolvability of ball structures 195

Proof of Proposition 2.2. It suffices to partition X = Y
⋃

Z so that Y is a
disjoint union of κ0 large subset and Z is κn-crowded for every n ∈ ω. We may
suppose that κ0 > 0. Choose α ∈ P such that |B(x,α)| ≥ 2κ0 for every x ∈ X.
By Zorn lemma, there exists a subset A ⊆ X such that {B(a,α) : a ∈ A} is
a maximal disjoint family. For every a ∈ A, partition B(a,α) = C(a)

⋃

D(a)
so that |C(a)| = κ0, |D(a)| ≥ κ0. Put Y =

⋃

a∈A
C(a), Z = X\Y and note

that Y can be partitioned in κ0-many large subsets. Fix n ∈ ω and choose
β ∈ P such that |B(x,β)| ≥ 2κn for every x ∈ X. Since B is multiplicative,
there exists γ ∈ P such that B(B(x,β),α) ⊆ B(x,γ) for every x ∈ X. Then
|B(z,γ)

⋂

Z| ≥ κn for every z ∈ Z and Z is κn-crowded. ✷

Proof of Theorem 2.3. Since cf B(X,d) = ℵ0, the first statement follows from
Proposition 2.1. The second statement follows from Proposition 2.2. ✷

To prove the next three theorems we use the filtrations of groups.
Let G be an infinite group with the identity e. A filtration of G is a family

{Gα : α < |G|} of subgroups of G such that
(i) G0 = {e}, G =

⋃

{Gα : α < |G|};
(ii) Gα ⊂ Gβ for all α < β < |G|;
(iii)

⋃

{Gα : α < β} = Gβ for every limit ordinal β;
(iv) Gα < |G| for every α < |G|.
Using a minimal well-ordering of G it is easy to construct a filtration of

G provided that G is not finitely generated. In particular, every uncountable
group admits a filtration.

For each α < |G|, decompose Gα+1 \ Gα to right cosets by Gα and fix some
set Xα of representatives so Gα+1 \ Gα = GαXα. Take an arbitrary element
g ∈ G, g 6= e and choose the smallest subgroup Gα with g ∈ Gα. By (iii),
α = α1 + 1 for some ordinal α1 < |G|. Hence, g ∈ Gα1+1 \ Gα1 and there exist
g1 ∈ Gα1, xα1 ∈ Xα1 such that g = g1xα1. If g1 6= e, we choose the ordinal
α2, the elements g2 ∈ Gα2+1 \ Gα2 and xα2 ∈ Xα2 such that g1 = g2xα2. Since
the set of ordinals < |G| is well-ordered, after finite number of steps we get the
representation

g = xαs(g)xαs(g)−1...xα2xα1, αs(g) < ... < α1,xαi ∈ Xαi.

Note that this representation is unique and put

γ1(g) = α1, γ2(g) = α2, ...,γs(g)(g) = αs(g), Γ(g) = {γ1(g), ...,γs(g)(g)}.

For every natural number n, denote

Dn = {g ∈ G : s(g) = n}.

Proof of Theorem 2.4. First suppose that |G| = κ. If G is countable, then
Bl(G,κ) is metrizable and we can apply Theorem 2.3. Assume that G is un-
countable and use the above filtration. For every α < |G|, put

Fα = {g ∈ G : γs(g)(g) = α}

and note that {Fα : α < G} is a pairwise disjoint family of left κ-large subsets.



196 I. V. Protasov

If κ < |G|, we choose a subgroup H of G with |H| = κ. By above paragraph,
there exists a partition P of H such that each subset P ∈ P is large in Bl(H,κ).
Decompose G to right cosets by H and fix some set X or representatives so
G = HX. Then {PX : P ∈ P} is a pairwise disjoint family of left κ-large
subsets of G. ✷

Proof of Theorem 2.5. If G is countable, then Bl(G,κ) is metrizable and we
have cores Bl(G,κ) = ℵ0. Suppose that G is uncountable and use the above
filtration. Observe that G \ {e} =

⋃

∞

n=1 Dn, fix a natural number n and show
that Dn is small in Bl(G,κ). Take an arbitrary subset F ∈ ℑ(G,κ). By
assumption, there exists β ∈ P such that F ⊆ Gβ so FDn ⊆ GβDn. Show that
G \ GβDn is left ℵ0-large. Choose the elements a1,a2, ...,an+1 of G such that

α1 ∈ Gβ+1 \ Gβ,a2 ∈ Gβ+2 \ Gβ+1, ...,an+1 ∈ Gβ+n+1 \ Gβ+n.

Take an arbitrary element g ∈ GβDn and put g = g0. If β + n ∈ Γ(g),
put ε0 = 0, otherwise ε0 = 1. Note that β + n ∈ Γ(a

ε0
n+1g0) and put g1 =

aε0n+1g0. If β + n − 1 ∈ Γ(g1), we put ε1 = 0, otherwise ε1 = 1. Note that
{β + n − 1,β + n} ⊆ Γ(aε1n+1g1) and put g2 = a

ε1
n g1. After n + 1 steps we get

{β,β + 1, ...,β + n} ⊆ Γ(aεn1 a
εn−1
2 ...a

ε0
n+1g).

It follows that (aεn1 a
εn−1
2 ...a

ε0
n+1g) /∈ GβDn. Put A = {e,a1,a2, ...,an+1},

K = An. We have shown that GβDn ⊆ K
−1(G\GβDn). Hence, G = K

−1(G\
GβDn) and G \ GβDn is left ℵ0-large. ✷

Proof of Theorem 2.6. If G is σ-compact, then cf Bl(G,C) = ℵ0 and Bl(G,C)
is metrizable. Hence, cores Bl(G,C) = ℵ0. Assume that G is not σ-compact.
Then we can easily construct a filtration {Gα : α < |G|} so that every subgroup
Gα, α > 0 is open. Repeat the arguments proving Theorem 2.5. ✷

Proof of Theorem 2.7. Two easy observations. A subset Y ⊆ X is large if and
only if Y ∈ ϕ. A subset Y ⊆ X is small if and only if X \ Y is large. ✷

4. Comments and open problems

Problem 4.1. Let B = (X,P,B) be a ball structure, κ be a cardinal such that
B is κ′-crowded for every κ′ < κ. Can X be partitioned in κ large subsets? By
Proposition 2.2, this is so if cfκ = ℵ0.

Problem 4.2. Let G be an infinite group, κ be an infinite cardinal, κ ≤ |G|.
Can G be κ-partitioned so that each cell of the partition is left and right κ-large?
If κ = ℵ0, this is so [14].

Let G be an infinite amenable (in particular, Abelian) group, µ be a Banach
measure on G. Clearly, µ(A) > 0 for every left ℵ0-large subset A of G. It
follows, that res Bl(G,ℵ0) = ℵ0. On the other hand, every free group of
infinite rank κ can be partitioned in κ left ℵ0-large subsets [11].



Resolvability of ball structures 197

Problem 4.3. Let G be a free Abelian group of rank ℵ2. Can G be partitioned
in ℵ2 ℵ1-large subsets.

Problem 4.4. Let G be an infinite group, κ be an infinite cardinal, κ ≤ |G|.
Can G be partitioned in ℵ0 left κ-small subsets? By Theorem 2.5, this is so if
|G| is a regular cardinal.

A topological space is called irresolvable if it can not be partitioned in two
dense subsets. Let us say that a ball structure B is irresolvable if resB = 1. By
Proposition 2.1, B is irresolvable if and only if crB = 1.

A topological space X is called κ-extraresolvable if X admits a family ℑ,
|ℑ| = κ of dense subsets such that F1

⋂

F2 is nowhere dense for all dis-
tinct subsets F1,F2 ∈ ℑ. It is important to remark that if 1 < κ < ω
then κ-extraresolvability is equivalent to κ-resolvability. The concept of κ-
extraresolvability was introduced by V. I. Malykhin [8]. As the referee pointed
out, the published paper where this concept appears for the first time in the
literature is [6].

Let us say that a ball structure B = (X,P,B) is κ-extraresolvable if X admits
a family ℑ, |ℑ| = κ of large subsets such that F1

⋂

F2 is small for all distinct
subsets F1,F2 ∈ ℑ.

If B is unbounded and κ-crowded, then there exists a family ℑ, |ℑ| = κℵ0 of
large subset of X such that F1

⋂

F2 is finite for all distinct subsets F1,F2 ∈ ℑ, so
B is κℵ0-extraresolvable. The extraresolvability of B is the cardinal sup {κ : B
is κ-extraresolvable}.

Problem 4.5. Determine (or evaluate) extraresolvability of ball structures of
metric spaces and groups.

The referee asked ”what about the infinite case in (iii) of Proposition 2.1?”
Let k be an uncountable ordinal and let G be a free group of rank k. By [13],
Bl(G,ℵ0) can be partitioned in k-many large subsets, but cr Bl(G,ℵ0) = ℵ0.
On the other hand, the statement (iii) remains true for k = ℵ0.

Acknowledgements. I would like to thank the referee for the long list of
corrections and suggestions to the previous version of the paper.

References

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groups, Annals of New York Acad. of Sciences, 767 (1995), 17-27.
[4] W. W. Comfort and S. Garcia-Ferreira, Resolvability: a selective survey and some new

results, Topology Appl. 74 (1996), 149-167.



198 I. V. Protasov

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Received March 2003

Accepted February 2004

I. V. Protasov (kseniya@profit.net.ua)
Department of Cybernetics, Kyiv University, Volodimirska 64, Kiev 01033,
UKRAINE