Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
INFTY Final Conference 2014
Strong measure zero in separable metric spaces
and Polish groups
Michael Hrusak
joint with W. Wohofsky and O. Zindulka
CCM-Morelia
Universidad Nacional Autónoma de México
[email protected]
Bonn
March 2014
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Contents
1
Uniform properties of metric spaces
2
Strong measure zero in Polish groups
3
The small ball property
4
Uniformity of Smz
5
Transitive covering in Polish groups
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Monotone metric spaces
Definition (Zindulka - 2005)
A metric space (X , d) is said to be monotone if there is a linear order <
on X and a c > 0 such that if x < y < z then d(x, y ) 6 c · d(x, z). The
space is σ-monotone if it it is a countable union of monotone subspaces.
Used in real analysis (Keleti-Máthé-Zindulka - 2013) to analyze
Lipschitz pre-images of n-dimensional cubes.
Has close connection to porosity ideals on metric spaces.
Question (H.-Zindulka - 2012)
Is there a metric space of size ℵ1 which is not σ-monotone?
(H.-Zindulka) Consistently there is no separable example. Also,
consistent with MA(σ-centered) that there is an example.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Strong measure zero
Definition ( Borel - 1919)
A metric space X has strong measure zero (Smz) if for any sequence
hεn : n ∈ ωi of positive numbers there is a cover {Un : n ∈ ω} of X such
that diam Un 6 εn for all n.
No perfect set is Smz.
Borel Conjecture: Every Smz set of reals (equivalently, of separable
metric space) is countable.
(Sierpiński - 1928) Consistently NO.
(Laver - 1976) Consistently YES.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The small ball property and uniformly meager spaces
Definition (Behrends and Kadets - 2001)
A metric space (X , d) has the small ball property (sbp) if for any ε > 0
there is a sequence hB(xn , εn ) : n ∈ ωi of balls with ε > εn and
limn→∞ εn = 0 that cover X .
Definition
A set A in a metric space (X , d) is uniformly nowhere dense if for any
sequence ε > 0 there is a δ > 0 such that for any x ∈ X there is a y ∈ X
such that B(y , δ) ⊆ B(x, ε) \ A. A set A ⊆ X is uniformly meager if it is
a countable union of uniformly nowhere dense sets.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Contents
1
Uniform properties of metric spaces
2
Strong measure zero in Polish groups
3
The small ball property
4
Uniformity of Smz
5
Transitive covering in Polish groups
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Smz and translates of meager sets
Proposition (Prikry - 197?)
Let X ⊆ R be such that X + M 6= R for all M ∈ M. Then X has strong
measure zero.
• (Prikry) Is the reverse true?
Theorem (Galvin-Mycielski-Solovay - 1976)
A set X ⊆ R has Smz if and only if X + M 6= R for all M ∈ M.
Extended by Fremlin and Kysiak (independently?):
Proposition (Fremlin and Kysiak - 200?)
Let G be a separable locally compact metric group. A set S ⊆ G has
strong measure zero if and only if S · M 6= G for all M ∈ M(G).
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Smz and translates of meager sets
Proposition (Prikry)
Let G be a separable group equipped with a left-invariant metric d and
let S ⊆ G be such that S · M 6= G for all M ∈ M(G). Then S has strong
measure zero with respect to d.
A slightly different reverse:
Proposition (HWZ)
Let G be a Polish group which is either locally compact, or equipped with
a complete invariant metric d, and let S ⊆ G be Smz with respect to d.
Then S · M 6= G for every uniformly meager M ⊆ G.
This gives the Fremlin-Kysiak result as: In a locally compact space
every meager set is uniformly meager.
Is the proposition true for all Polish groups equipped with a
complete left-invariant metric?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Smz and translates of meager sets
Natural questions
1
Is the Galvin-Mycielski-Solovay/Fremlin-Kysiak result true for all
Polish groups?
2
Is the Prikry result true with meager replaced by uniformly meager?
• Note that assuming the Borel Conjecture the first item is trivially true.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The Baer-Specker group Zω
Theorem (CH)
There is a set X ⊆ Zω such that X + M 6= Zω for every uniformly
meager M, and yet X ∈
/ Smz(Zω ).
Theorem (CH)
There is a Smz set X ⊆ Zω such that X + M 6= Zω for every
M ∈ M(Zω ).
The question
Is the Galvin-Mycielski-Solovay/Fremlin-Kysiak result true in general only
for locally compact Polish groups?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The Baer-Specker group Zω
Theorem (CH)
There is a set X ⊆ Zω such that X + M 6= Zω for every uniformly
meager M, and yet X ∈
/ Smz(Zω ).
Theorem (CH)
There is a Smz set X ⊆ Zω such that X + M 6= Zω for every
M ∈ M(Zω ).
The question
Is the Galvin-Mycielski-Solovay/Fremlin-Kysiak result true in general only
for locally compact Polish groups?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Contents
1
Uniform properties of metric spaces
2
Strong measure zero in Polish groups
3
The small ball property
4
Uniformity of Smz
5
Transitive covering in Polish groups
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The small ball property
Definition (Behrends and Kadets - 2001)
A metric space (X , d) has the small ball property (sbp) if for any ε > 0
there is a sequence hB(xn , εn ) : n ∈ ωi of balls with ε > εn and
limn→∞ εn = 0 that cover X .
sbp is a σ-additive property.
Every (σ-) compact space has sbp.
Every Smz space has sbp.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The small ball property
A right-continuous, nondecreasing function g : (0, ∞) → (0, ∞) with
limr →0 g (r ) = 0 is called a gauge. Given a gauge g and Y ⊆ X , the
g -dimensional Hausdorff measure of Y is defined thus:
nX
o
[
Hg (Y ) = sup inf
g (diam E ) : Y ⊆
E ∧ ∀E ∈ E (diam(E ) 6 δ) .
δ>0
E ∈E
Proposition (HWZ)
The following are equivalent.
(i) X has sbp, and diam En → 0,
(ii) X admits a base {Bn : n ∈ ω} such that diam Bn → 0,
(iii) for every sequence hεm : m ∈ ωi of positive
reals there is a sequence
S
hFm : m ∈ ωi of finite sets such that m∈ω {B(x, εm ) : x ∈ Fm }
covers X ,
(iv) there is a gauge g such that Hg (X ) = 0.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The small ball property
Proposition (HWZ)
The following are equivalent.
(i) X has sbp,
(ii) there is a gauge g such that Hg (X ) = 0
Compare to:
Proposition (Besicovitch - 1933)
The following are equivalent.
(i) X has Smz,
(ii) Hg (X ) = 0 for every gauge g .
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
The small ball property
In a similar vein:
Proposition (HWZ)
A metrizable space X has the Menger Property iff X has sbp in all
compatible metrics.
Compared to:
Proposition (Fremlin-Miller - 1988)
A metrizable space X has the Rothberger Property iff X has Smz in all
compatible metrics.
A space X has the Menger Property if for every sequence
hUn : n ∈ ωi of
S
open covers there are finite sets Fn ⊆ Un such that n∈ω Fn covers X ,
while X has the Rothberger Property if for every sequence hUn : n ∈ ωi
of open covers there are Un ∈ Un such that {Un : n ∈ ω} covers X .
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Cardinal invariants of sbp
Proposition (HWZ)
If a metric space X does not have sbp, then
(i) non(sbp(X )) = cof(sbp(X )) = d,
(ii) add(sbp(X )) = cov(sbp(X )) = b.
Proposition (HWZ)
For a subset X ⊆ ω ω , the following are equivalent:
(i) X has sbp,
(ii) no isometric copy of X in ω ω is dominating,
(iii) no uniformly continuous image of X in ω ω is dominating.
Question.
Is Borel(ω ω )/ sbp forcing equivalent to the Laver forcing?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Contents
1
Uniform properties of metric spaces
2
Strong measure zero in Polish groups
3
The small ball property
4
Uniformity of Smz
5
Transitive covering in Polish groups
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Related cardinal invariants
Definition
eq = min{|F | : F is a bounded family,
∀g ∈ ω ω ∃f ∈ F ∀n ∈ ω f (n) 6= g (n)}.
ed = min{|F | : F is a bounded family of infinite partial functions,
∀g ∈ ω ω ∃f ∈ F ∀n ∈ dom f f (n) 6= g (n)}.
• (Bartoszyński - 1987) omitting “bounded” in the definition of eq gives
cov(M).
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Related cardinal invariants
•
•
•
•
•
bO
/d
O
add(M)
/ cov(M)
/ ed
/ eq
/ non(N )
add(M) = min{b, eq} = min{b, ed}
cov(M) = min{d, ed},
(Goldstern-Judah-Shelah - 1993) Con(cov(M) < min{d, eq})
(Fremlin-Miller - 1988) non(Smz(ω ω )) = cov(M)
(Bartoszyński-Judah - 1995) non(Smz(2ω )) = eq
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Uniformity of Smz
Theorem (HWZ)
Let X be a separable metric space. Then:
1
If X is σ-totally bounded then eq 6 non(Smz(X )),
2
if X is sbp then ed 6 non(Smz(X )), and
3
if X is not sbp then cov(M) 6 non(Smz(X )) 6 d.
Theorem (HWZ)
Let X be an uncountable analytic separable metric space. Then:
1
If X is σ-totally bounded then non(Smz(X )) = eq,
2
if X is sbp then ed 6 non(Smz(X )) 6 eq, and
3
if X is not sbp then cov(M) 6 non(Smz(X )) 6 min{eq, d}.
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Uniformity of Smz in Polish groups
There is a Hurewicz type dichotomy:
Proposition (HWZ)
A Polish group equipped with a complete, left-invariant metric is either
locally compact, or else contains a uniform copy of ω ω and, in particular,
is not sbp.
Corollary (HWZ)
Let G be a Polish group equipped with a complete, left-invariant metric.
1
If G is locally compact, then non(Smz(G)) = eq,
2
if G is not locally compact, then non(Smz(G)) = cov(M).
Question
Is the corollary true for all Polish groups?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Consistency results
Theorem (Szpilrajn - 1935)
If X is a separable metric which is not of universal measure zero, then
non(Smz(X )) 6 non(N ).
Recall that a metric space X is of universal measure zero if there is no
probability Borel measure on X vanishing on singletons.
Theorem (HWZ)
It is consistent that there is a non-sbp set X ⊆ ω ω such that
d = non(Smz(X )) > non(N ) = cov(M).
Question
Is cof(N ) an upper bound for non(Smz(X )) for any non-Smz separable
space X ?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Consistency results
Theorem (HWZ)
It is consistent that there is an sbp set X ⊆ ω ω such that
non(Smz(X )) = ed < eq.
Question
Is non(Smz(X )) = eq for all sbp analytic (Borel, absolutely Gδ ,. . . )
spaces X ? Is non(Smz(X )) = cov(M) for all non-sbp analytic (Borel,
absolutely Gδ ,. . . ) spaces X ?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Contents
1
Uniform properties of metric spaces
2
Strong measure zero in Polish groups
3
The small ball property
4
Uniformity of Smz
5
Transitive covering in Polish groups
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Transitive covering
Definition (Miller-Steprāns - 2006)
Given a Polish group G, let cov∗G be the minimal cardinality of a set
A ⊆ G such that A · M = G for some meager set M ⊆ G.
• Related to Smz via the
Prikry/Galvin-Mycielski-Solovay/Fremlin-Kysiak results.
Questions (Miller-Steprāns - 2006)
Is it consistent to have a compact Polish group G such that
cov∗G > eq?
Is it true that cov∗G > eq for any infinite compact Polish group G?
Is it true that for every non-discrete Polish group G either cov∗G = eq
or cov∗G = cov(M)?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Transitive covering
Theorem (HWZ)
If G is a Polish group, then cov(M) 6 cov∗G 6 eq.
Theorem (HWZ)
If G is a non-discrete, locally compact Polish group, then cov∗G = eq.
Theorem (HWZ)
Let G be a Polish group endowed with a complete left-invariant metric. If
G is not locally compact, then cov∗G = cov(M).
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Transitive covering - Questions
Question
Is there consistently a non-locally compact Polish group G such that
cov∗G > cov(M)?
Question
Is there consistently a Polish group G such that non(Smz(G)) 6= cov∗G ?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups
Uniform properties of metric spaces
Strong measure zero in Polish groups
The small ball property
Uniformity of Smz
Transitive covering in Polish groups
Thank you for your attention!
Have you found a non-σ-monotone metric space of size ℵ1 yet?
M. Hrusak
Strong measure zero in separable metric spaces and Polish groups