A critical ideal with respect to Mazurkiewicz’s
theorem in terms of the Katětov order
Rafal Filipów
University of Gdańsk
Katětov order
Definition (Katětov order)
I ≤K J ⇐⇒ ∃f : N → N ∀A (A ∈ I ⇒ f −1 [A] ∈ J )
Katětov order was introduced by Miroslav Katětov in 1968 to
study convergence in topological spaces.
(David Meza Alcántara; Nikodem Mrożek) There is an order
embedding of P(ω)/FIN into Borel ideals ordered by the
Katětov order
More on Katětove order can be found in Hrušak survey
“Combinatorics of filters and ideals”
Critical ideals in the Katětov order
Let PROPERTY(I) be a property of ideals. For instance
1
I is a P-ideal;
2
I is weakly Ramsey;
3
I is a Polish ideal of worse sort.
Definition
An ideal C is critical for PROPERTY(I) in terms of the Katětov
order ≤K iff
PROPERTY(I) ⇐⇒ C ≤K I
Example: P-points
Definition
An ideal I is a P-ideal if for any countable family A ⊂ I there is
B ∈ I such that A \ B is finite for every A ∈ A
Definition
FIN ⊗ FIN is an ideal on N × N defined by
A ∈ FIN ⊗ FIN ⇐⇒ ∃n ∈ N ∀k > n ({m : (k, m) ∈ A} is finite)
Theorem (Folklore)
Let I be a maximal ideal.
I is a P-ideal ⇐⇒ FIN ⊗ FIN 6≤K I
Example: ideal limits of seq. of continuous functions
Definition
A seq. (xn ) is I-convergent to x if {n : xn ∈
/ U} ∈ I for every
open nbhd. U of x.
A seq. fn : X → R is I-pointwise convergent to f : X → R if
(fn (x)) is I-convergent to f (x) for every x.
B1I - family of all I-pointwise limit of seq. of continuous fun.
Theorem (Folklore)
If I is a maximal ideal, then B1I contains a nonmeasurable function
Theorem (Katětov, Debs-Saint Raymond, Laczkovich-Reclaw)
Let I be a Borel ideal.
B1I = B1 ⇐⇒ FIN ⊗ FIN 6≤K I
Example: ideal e-limits of seq. of quasi-continuous fun.
Definition (Császár-Laczkovich)
A seq. fn : X → R is e-convergent to f : X → R if there is a
seq. of positive reals εn → 0 such that {n : |fn (x) − f (x)| < εn } for
almost all n.
Definition (Kempisty)
A fun. f : X → R is quasi-continuous at x0 if for every ε > 0 and
every open nbhd U of x0 there is nonempty open G ⊂ U such that
|f (x) − f (x0 )| < ε for every x ∈ G .
Theorem (Kwela-Staniszewski 2015)
Let I be a Borel ideal.
B1e−I = B1e ⇐⇒ WR 6≤K I
Bolzano-Weierstrass in the realm of functions
Theorem (Bolzano-Weierstrass)
Every bounded sequences of reals has a convergent subsequence.
Theorem (Folklore)
There is a bounded sequence of functions fn : R → R without
pointwise convergent subsequence.
Question (Saks)
Let fn : R → R be a bounded sequence of functions. Is there a
subsequence (fkn ) such that (fkn (x)) is convergent for uncountably
many x?
Theorem
CH ⇒ No (Sierpiński)
MA+¬CH ⇒ Yes (Fuchino-Plewik)
B-W in the realm of continuous functions
Theorem (Folklore)
There is a bounded sequence of continuous functions fn : R → R
without pointwise convergent subsequence.
Theorem (Mazurkiewicz)
If (fn ) is a bounded sequence of continuous functions then there
exists a subsequence (fkn ) and a perfect set X ⊂ R such that
(fkn (x)) is convergent for every x ∈ X .
Remark
Cichoń and Żeberski found (the hard way) proof of Mazurkiewicz
theorem with the aid of Shoenfield absolutness theorem.
Ideal version of Mazurkiewicz’s theorem
Theorem (Mazurkiewicz)
If (fn ) is a bounded sequence of continuous functions then there
exists a subsequence (fkn ) and a perfect set X ⊂ R such that
(fkn (x)) is convergent for every x ∈ X .
Definition
CONV is an ideal on Q ∩ [0, 1] generated by sequences in Q ∩ [0, 1]
convergent in [0, 1].
Theorem (Filipów-Mrożek-Reclaw-Szuca)
Let I be an analytic P-ideal. TFAE:
For every bounded sequence (fn ) of continuous functions there
exists A ∈
/ I and a perfect set X ⊂ R such that (fn (x))n∈A is
convergent for every x ∈ X .
CONV 6≤K I
Ideal version of Mazurkiewicz’s theorem - main steps of
the proof
Step 1 (Meza-Alcantara)
CONV 6≤K I ⇐⇒ I has finBW property
Step 2
Let I be an analytic P-ideal. TFAE:
I has finBW property
The submeasure || · ||φ is not nonatomic
The ideal I can be extended to an Fσ ideal
Step 3
Let I be an Fσ ideal. For every bounded sequence (fn ) of
continuous functions there exists A ∈
/ I and a perfect set X ⊂ R
such that the subsequence (fn X )n∈A is pointwise convergent.
Question of Hrušák
Question (Hrušák)
Let I be a Borel ideal. Are the following conditions equivalent?
CONV 6≤K I
I can be extended to Fσ ideal
Theorem (Filipów-Mrożek-Reclaw-Szuca)
Let I be an analytic P-ideal. TFAE:
CONV 6≤K I
I has finBW property
The submeasure || · ||φ is not nonatomic
I can be extended to an Fσ ideal