Ideal quasi-normal convergence and related notions y , J . - - PowerPoint PPT Presentation

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Ideal quasi-normal convergence and related notions

Pratulananda Das∗, L.Bukovsk´ y, J.ˇ Supina

* Department of Mathematics, Jadavpur University, West Bengal

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Basic notions

Ideal: A hereditary family I ⊆ P(ω) (B ∈ I for any B ⊆ A ∈ I) that is closed under unions (A ∪ B ∈ I for any A, B ∈ I), contains all finite subsets of ω and ω ∈ I.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Basic notions

Ideal: A hereditary family I ⊆ P(ω) (B ∈ I for any B ⊆ A ∈ I) that is closed under unions (A ∪ B ∈ I for any A, B ∈ I), contains all finite subsets of ω and ω ∈ I. Filter: For A ⊆ P(ω) we denote Ad = {ω \ A : A ∈ A}. A family F ⊆ P(ω) is called a filter if Fd is an ideal.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Basic notions

Ideal: A hereditary family I ⊆ P(ω) (B ∈ I for any B ⊆ A ∈ I) that is closed under unions (A ∪ B ∈ I for any A, B ∈ I), contains all finite subsets of ω and ω ∈ I. Filter: For A ⊆ P(ω) we denote Ad = {ω \ A : A ∈ A}. A family F ⊆ P(ω) is called a filter if Fd is an ideal. Associated Filter: If I is a proper ideal in Y (i.e. Y / ∈ I, I = {∅}), then the family of sets F(I) = {M ⊂ Y : there exists A ∈ I : M = Y \ A} is a filter in Y.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Basic notions

Ideal: A hereditary family I ⊆ P(ω) (B ∈ I for any B ⊆ A ∈ I) that is closed under unions (A ∪ B ∈ I for any A, B ∈ I), contains all finite subsets of ω and ω ∈ I. Filter: For A ⊆ P(ω) we denote Ad = {ω \ A : A ∈ A}. A family F ⊆ P(ω) is called a filter if Fd is an ideal. Associated Filter: If I is a proper ideal in Y (i.e. Y / ∈ I, I = {∅}), then the family of sets F(I) = {M ⊂ Y : there exists A ∈ I : M = Y \ A} is a filter in Y. It is called the filter associated with the ideal I.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Basic notions

Ideal: A hereditary family I ⊆ P(ω) (B ∈ I for any B ⊆ A ∈ I) that is closed under unions (A ∪ B ∈ I for any A, B ∈ I), contains all finite subsets of ω and ω ∈ I. Filter: For A ⊆ P(ω) we denote Ad = {ω \ A : A ∈ A}. A family F ⊆ P(ω) is called a filter if Fd is an ideal. Associated Filter: If I is a proper ideal in Y (i.e. Y / ∈ I, I = {∅}), then the family of sets F(I) = {M ⊂ Y : there exists A ∈ I : M = Y \ A} is a filter in Y. It is called the filter associated with the ideal I.

• If I ⊆ P(ω) is an ideal then B ⊆ I is a base of I if for any

A ∈ I there is B ∈ B such that A ⊆ B. We recall a folklore fact: the family of all finite intersections of elements of a family A ⊆ [ω]ω is a base of some filter if and only if A has the finite intersection property, shortly f.i.p..

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

cof(I): For an ideal I we denote cof(I) = min{|A| : A ⊆ I ∧ A is a base of I}.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

cof(I): For an ideal I we denote cof(I) = min{|A| : A ⊆ I ∧ A is a base of I}. almost contained: A set A is almost contained in a set B, written A ⊆∗ B, if A \ B is finite. Assume that A ⊆ I is such that every B ∈ I is almost contained in some A ∈ A. Then B = {A ∪ F : A ∈ A ∧ F ∈ [ω]<ω} is a base of I. Moreover, if A is infinite, then |B| = |A|.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

cof(I): For an ideal I we denote cof(I) = min{|A| : A ⊆ I ∧ A is a base of I}. almost contained: A set A is almost contained in a set B, written A ⊆∗ B, if A \ B is finite. Assume that A ⊆ I is such that every B ∈ I is almost contained in some A ∈ A. Then B = {A ∪ F : A ∈ A ∧ F ∈ [ω]<ω} is a base of I. Moreover, if A is infinite, then |B| = |A|. P-ideal: An ideal I is said to be a P-ideal, if for any countable A ⊆ I there exists a set B ∈ I such that A ⊆∗ B for each A ∈ A. Some authors say that I satisfies the property (AP). If A ⊆ ω is such that ω \ A is infinite, then A∗ = {B ⊆ ω : B ⊆∗ A} is a P-ideal with a countable base.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

pseudointersection: An infinite set B ⊆ ω is said to be a pseudointersection of a family A ⊆ [ω]ω if B ⊆∗ A for any A ∈ A. We can introduce the dual notion: a set B is a pseudounion of the family A if ω \ B is infinite and if A ⊆∗ B for any A ∈ A. Thus an ideal I is P-ideal if and only if every countable subfamily of I has a pseudounion belonging to I. If a pseudounion A of I belongs to I, then I = A∗.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

pseudointersection: An infinite set B ⊆ ω is said to be a pseudointersection of a family A ⊆ [ω]ω if B ⊆∗ A for any A ∈ A. We can introduce the dual notion: a set B is a pseudounion of the family A if ω \ B is infinite and if A ⊆∗ B for any A ∈ A. Thus an ideal I is P-ideal if and only if every countable subfamily of I has a pseudounion belonging to I. If a pseudounion A of I belongs to I, then I = A∗. Tall ideal: An ideal I is tall, if for any B ∈ [ω]ω, there exists an A ∈ I such that A ∩ B is infinite. Thus, an ideal I has a pseudounion if and only if I is not tall.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

pseudointersection number: The pseudointersection number is the cardinal p = min{|A| : (A ⊆ [ω]ω has f.i.p. and has no pseudointersection) Thus, if I is an ideal with cof(I) < p, then I has a pseudounion. Since p > ℵ0, any ideal with a countable base has a pseudounion.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

pseudointersection number: The pseudointersection number is the cardinal p = min{|A| : (A ⊆ [ω]ω has f.i.p. and has no pseudointersection) Thus, if I is an ideal with cof(I) < p, then I has a pseudounion. Since p > ℵ0, any ideal with a countable base has a pseudounion.

• An ideal I with a countable base can be constructed with

a pseudounion such that no pseudounion of I belongs to I and such that I is not a P-ideal. Assuming p > ℵ1, one can construct a P-ideal I with an uncountable base of cardinality < p such that no pseudounion of I belongs to I.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Ideal convergence: A sequence xn : n ∈ ω of elements

• f a topological space X I-converges to x ∈ X, written

xn

I

− → x, if for each neighborhood U of x, the set {n ∈ ω : xn / ∈ U} ∈ I, i.e., if the function xn : n ∈ ω from ω into X converges modulo filter Id to x in the sense of

• H. Cartan.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Ideal convergence: A sequence xn : n ∈ ω of elements

• f a topological space X I-converges to x ∈ X, written

xn

I

− → x, if for each neighborhood U of x, the set {n ∈ ω : xn / ∈ U} ∈ I, i.e., if the function xn : n ∈ ω from ω into X converges modulo filter Id to x in the sense of

• H. Cartan.

Ideal divergence: A sequence xn : n ∈ ω is I-divergent to ∞, written xn

I

− → ∞, if {n : xn < a} ∈ I for any positive real a > 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Ideal convergence: A sequence xn : n ∈ ω of elements

• f a topological space X I-converges to x ∈ X, written

xn

I

− → x, if for each neighborhood U of x, the set {n ∈ ω : xn / ∈ U} ∈ I, i.e., if the function xn : n ∈ ω from ω into X converges modulo filter Id to x in the sense of

• H. Cartan.

Ideal divergence: A sequence xn : n ∈ ω is I-divergent to ∞, written xn

I

− → ∞, if {n : xn < a} ∈ I for any positive real a > 0.

• By function f, we always mean a real function defined
• n X. A sequence of real functions fn : n ∈ ω

I-converges to a real function f on X, written fn

I

− → f, if fn(x)

I

− → f(x) for each x ∈ X.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I- Quasinormal convergence: A sequence of real functions fn : n ∈ ω on X I-quasi-normally converges to a real function f on X, shortly fn

IQN

− → f on X, if there exists a sequence of reals εn : n ∈ ω that I-converges to 0 (the control sequence) and such that {n ∈ ω : |fn(x) − f(x)| ≥ εn} ∈ I for any x ∈ X.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I- Quasinormal convergence: A sequence of real functions fn : n ∈ ω on X I-quasi-normally converges to a real function f on X, shortly fn

IQN

− → f on X, if there exists a sequence of reals εn : n ∈ ω that I-converges to 0 (the control sequence) and such that {n ∈ ω : |fn(x) − f(x)| ≥ εn} ∈ I for any x ∈ X. strongly I-quasi-normal convergence: We say that a sequence strongly I-quasi-normally converges to f if the control sequence is 2−n : n ∈ ω. We write fn

sIQN

− → f. In fact one can replace the sequence 2−n : n ∈ ω by any sequence εn : n ∈ ω of positive reals such that ∞

n=0 εn < ∞.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I - Uniform Convergence A sequence of real functions fn : n ∈ ω on X I-uniformly converges to a real function f, shortly fn

I-u

− → f, if there exists a set A ∈ I such that {n ∈ ω : |fn(x) − f(x)| ≥ ε} ⊆ A for any ε > 0 and any x ∈ X.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I - Uniform Convergence A sequence of real functions fn : n ∈ ω on X I-uniformly converges to a real function f, shortly fn

I-u

− → f, if there exists a set A ∈ I such that {n ∈ ω : |fn(x) − f(x)| ≥ ε} ⊆ A for any ε > 0 and any x ∈ X.

• Evidently the notion of I - Uniform Convergence is

stronger than the notion of I- Quasinormal convergence which is again stronger than the notion of I- pointwise

• convergence. Examples have been constructed in this

respect.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 1

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 1

Theorem The following are equivalent (i) cof(I) = κ. (ii) For any set X and for any sequence of real functions, if fn

IQN

− → f on X, then there exist sets Xξ, ξ < κ such that X =

ξ<κ Xξ and fn I-u

− → f on each Xξ. (iii) For any set X ⊆ P(ω) and for any sequence of real functions, if fn

IQN

− → f on X, then there exit sets Xξ, ξ < κ such that X =

ξ<κ Xξ and fn I-u

− → f on each Xξ. Moreover, if X is a topological space and fn, n ∈ ω are continuous, then in both cases we can assume that the sets Xξ are closed.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 2

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 2

Theorem The following are equivalent: (i) The set C is a pseudounion of the ideal I. (ii) For every set X and every fn

IQN

− → f on X with the control εn : n ∈ ω, there exist sets Xk, k ∈ ω such that X =

• k

Xk and fn

C∗-u

− → f with same control εn : n ∈ ω on each Xk. (iii) For every set X and every fn

IQN

− → f on X with the control εn : n ∈ ω, there exists a cardinal (may be finite) κ and there exist sets Xξ, ξ < κ such that X =

• ξ<κ

Xξ and fn

C∗-u

− → f with same control εn : n ∈ ω on each Xξ.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Lemma 1

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Lemma 1

Assume that I ⊆ P(ω) is an ideal with a pseudounion C. Let A = ω \ C. Then a) For any sequence xn : n ∈ ω of reals, if xn

I

− → 0 then xeA(n) → 0. b) For any sequence fn : n ∈ ω of real functions defined on X, if fn

I

− → 0 on X, then feA(n) → 0 on X. c) For any sequence fn : n ∈ ω of real functions defined on X, if fn

IQN

− → 0 on X, then feA(n)

QN

− → 0 on X.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

(I,J )QN-space: A topological space X is an (I,J )QN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to 0 on X, we have fn

J QN

− → 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

(I,J )QN-space: A topological space X is an (I,J )QN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to 0 on X, we have fn

J QN

− → 0. (I,J )wQN-space: A topological space X is an (I,J )wQN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to zero on X, there exists a sequence mn : n ∈ ω such that fmn

J QN

− → 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

(I,J )QN-space: A topological space X is an (I,J )QN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to 0 on X, we have fn

J QN

− → 0. (I,J )wQN-space: A topological space X is an (I,J )wQN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to zero on X, there exists a sequence mn : n ∈ ω such that fmn

J QN

− → 0.

• We can take mn

J

− → ∞. Indeed, instead of fn : n ∈ ω, consider the sequence |fn| + 2−n : n ∈ ω. Then for any a > 0 and any x ∈ X we have {n : mn ≤ a} ⊆ {n : |fmn(x)| + 2−mn ≥ 2−a} ∈ J .

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

(I,J )QN-space: A topological space X is an (I,J )QN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to 0 on X, we have fn

J QN

− → 0. (I,J )wQN-space: A topological space X is an (I,J )wQN-space if for any sequence fn : n ∈ ω of continuous real functions I-converging to zero on X, there exists a sequence mn : n ∈ ω such that fmn

J QN

− → 0.

• We can take mn

J

− → ∞. Indeed, instead of fn : n ∈ ω, consider the sequence |fn| + 2−n : n ∈ ω. Then for any a > 0 and any x ∈ X we have {n : mn ≤ a} ⊆ {n : |fmn(x)| + 2−mn ≥ 2−a} ∈ J .

• If the sequence fn : n ∈ ω is decreasing we obtain the

notions of an (I,J )mQN-space and an (I,J )wmQN-space.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 3

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 3

Theorem Let I and J be ideals on ω. a) If I has a pseudounion, then every J wQN-space is an (I,J )wQN-space. b) If J has a pseudounion, then every J QN-space is a QN-space and every (I,J )wQN-space is an (I,Fin)wQN-space.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 3

Theorem Let I and J be ideals on ω. a) If I has a pseudounion, then every J wQN-space is an (I,J )wQN-space. b) If J has a pseudounion, then every J QN-space is a QN-space and every (I,J )wQN-space is an (I,Fin)wQN-space.

• Similar results hold true for (I,J )mQN-spaces and

(I,J )wmQN-spaces.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 3

Theorem Let I and J be ideals on ω. a) If I has a pseudounion, then every J wQN-space is an (I,J )wQN-space. b) If J has a pseudounion, then every J QN-space is a QN-space and every (I,J )wQN-space is an (I,Fin)wQN-space.

• Similar results hold true for (I,J )mQN-spaces and

(I,J )wmQN-spaces. Corollary If I ⊆ J and the ideal J has a pseudounion, then every (I,J )QN-space is a QN-space.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Problem For which ideal I not containing an isomorphic copy of Fin×Fin do we have IQN = QN? Similarly for IQN-, IwQN-, ImQN- and IwmQN-spaces.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Problem For which ideal I not containing an isomorphic copy of Fin×Fin do we have IQN = QN? Similarly for IQN-, IwQN-, ImQN- and IwmQN-spaces.

• J. ˇ

Supina has very recently showed that assuming p = c, for a γ-space X which is not a QN-space (the existence of such space was proved by Bukovski et al) there exists a tall ideal I, not containing an isomorphic copy of Fin×Fin, such that X is an IQN-space. We can even assume that I is a maximal ideal. Anyway, that is only very partial answer to our Problem 5.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

(α1): We recall that a topological space Y has the Arkhangel’skiˇ ı’s property (α1) if (α1) for any y ∈ Y and any sequence yn,m : m ∈ ω : n ∈ ω of sequences such that limm→∞ yn,m = y for each n, there exists a sequence zm : m ∈ ω such that limm→∞ zm = y and {yn,m : m ∈ ω} ⊆∗ {zm : m ∈ ω} for each n,

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

(α1): We recall that a topological space Y has the Arkhangel’skiˇ ı’s property (α1) if (α1) for any y ∈ Y and any sequence yn,m : m ∈ ω : n ∈ ω of sequences such that limm→∞ yn,m = y for each n, there exists a sequence zm : m ∈ ω such that limm→∞ zm = y and {yn,m : m ∈ ω} ⊆∗ {zm : m ∈ ω} for each n, (α4): Y has the Arkhangel’skiˇ ı’s property (α4) if (α4) for any y ∈ Y and any sequence yn,m : m ∈ ω : n ∈ ω of sequences such that limm→∞ yn,m = y for each n, there exists a sequence mn : n ∈ ω such that limn→∞ yn,mn = y.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

For ideals I and J we can modify the properties (α1) and (α4) for the space Cp(X) (or any space of real functions): (I, J -α1) If a sequence fn,m : m ∈ ω : n ∈ ω of sequences of continuous real functions is such that fn,m

I

− → 0 for each n, then there exists a sequence Bn : n ∈ ω ⊆ J ,

n∈ω Bn = ω, such that

(∀ε > 0)(∀x ∈ X)(∃A ∈ J )(∀n, m)(m ∈ A ∪ Bn → |fn,m(x)| < ε). (1)

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

For ideals I and J we can modify the properties (α1) and (α4) for the space Cp(X) (or any space of real functions): (I, J -α1) If a sequence fn,m : m ∈ ω : n ∈ ω of sequences of continuous real functions is such that fn,m

I

− → 0 for each n, then there exists a sequence Bn : n ∈ ω ⊆ J ,

n∈ω Bn = ω, such that

(∀ε > 0)(∀x ∈ X)(∃A ∈ J )(∀n, m)(m ∈ A ∪ Bn → |fn,m(x)| < ε). (1) and (I, J -α4) If a sequence fn,m : m ∈ ω : n ∈ ω of sequences of continuous real functions is such that fn,m

I

− → 0 for each n, then there exists a sequence mn : n ∈ ω such that fn,mn

J

− → 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 4 and 5

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 4 and 5

Theorem If X is a topological space then the following are equivalent: (i) X is an (I, sJ )wQN-space. (ii) Cp(X) has the property (I, J -α4).

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 4 and 5

Theorem If X is a topological space then the following are equivalent: (i) X is an (I, sJ )wQN-space. (ii) Cp(X) has the property (I, J -α4). Theorem For any topological space X and any ideal I, the following are equivalent. (i) X is an (I,J )QN-space. (ii) Cp(X) possesses the property (I, J -α1).

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

We can also introduce the ideal convergence modifications of properties (α0) and (α0)∗ for the space Cp(X), which were introduced by Bukovski and Hales (2007): (I, J -α0) If a sequence fn,m : m ∈ ω : n ∈ ω of sequences of continuous real functions is such that fn,m

I

− → 0 for each n, then there exists a J -diverging to ∞ sequence nm : m ∈ ω such that fnm,m

J QN

− → 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

We can also introduce the ideal convergence modifications of properties (α0) and (α0)∗ for the space Cp(X), which were introduced by Bukovski and Hales (2007): (I, J -α0) If a sequence fn,m : m ∈ ω : n ∈ ω of sequences of continuous real functions is such that fn,m

I

− → 0 for each n, then there exists a J -diverging to ∞ sequence nm : m ∈ ω such that fnm,m

J QN

− → 0. and (I, J -α∗

0) If a sequence fn : n ∈ ω pointwise

converges to 0 and a sequence fn,m : m ∈ ω : n ∈ ω

• f sequences of continuous real functions is such that

fn,m

I

− → fn for each n, then there exists a J -diverging to ∞ sequence nm : m ∈ ω such that fnm,m

J QN

− → 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

We can also introduce the ideal convergence modifications of properties (α0) and (α0)∗ for the space Cp(X), which were introduced by Bukovski and Hales (2007): (I, J -α0) If a sequence fn,m : m ∈ ω : n ∈ ω of sequences of continuous real functions is such that fn,m

I

− → 0 for each n, then there exists a J -diverging to ∞ sequence nm : m ∈ ω such that fnm,m

J QN

− → 0. and (I, J -α∗

0) If a sequence fn : n ∈ ω pointwise

converges to 0 and a sequence fn,m : m ∈ ω : n ∈ ω

• f sequences of continuous real functions is such that

fn,m

I

− → fn for each n, then there exists a J -diverging to ∞ sequence nm : m ∈ ω such that fnm,m

J QN

− → 0.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 5

Theorem For a topological space X the following are equivalent. (i) X is an (I, J )QN-space. (ii) Cp(X) has the property (I, J -α0). (iii) Cp(X) has the property (I, J -α∗

0).

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I-γ-cover: Let I be an ideal. A sequence Un : n ∈ ω of subsets of a topological space X is said to be an I-γ-cover, if for every n, Un = X, and for every x ∈ X, the set {n ∈ ω : x / ∈ Un} belongs to I.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I-γ-cover: Let I be an ideal. A sequence Un : n ∈ ω of subsets of a topological space X is said to be an I-γ-cover, if for every n, Un = X, and for every x ∈ X, the set {n ∈ ω : x / ∈ Un} belongs to I.

• We shall identify a countable γ-cover with a Fin-γ-cover.

One can easily observe that in this case the enumeration is

• inessential. The family of all open I-γ-covers of a given

topological space X will be denoted by I-Γ.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

I-γ-cover: Let I be an ideal. A sequence Un : n ∈ ω of subsets of a topological space X is said to be an I-γ-cover, if for every n, Un = X, and for every x ∈ X, the set {n ∈ ω : x / ∈ Un} belongs to I.

• We shall identify a countable γ-cover with a Fin-γ-cover.

One can easily observe that in this case the enumeration is

• inessential. The family of all open I-γ-covers of a given

topological space X will be denoted by I-Γ.

• A cover Vn : n ∈ ω is called a refinement of the cover

Un : n ∈ ω if Vn ⊆ Un for each n ∈ ω. An I-γ-cover Un : n ∈ ω is shrinkable if there exists a closed I-γ-cover that is a refinement of Un : n ∈ ω. We denote by I-Γsh the family of all open shrinkable I-γ-covers.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

• If Un : n ∈ ω and Vn : n ∈ ω are I-γ-covers, then

Un ∩ Vn : n ∈ ω is an I-γ-cover. If Un : n ∈ ω and Vn : n ∈ ω are shrinkable I-γ-covers, then Un ∩ Vn : n ∈ ω is a shrinkable I-γ-cover. Finally, if Un : n ∈ ω is an I-γ-cover and Un ⊆ Vn for each n, then Vn : n ∈ ω is an I-γ-cover.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

• If Un : n ∈ ω and Vn : n ∈ ω are I-γ-covers, then

Un ∩ Vn : n ∈ ω is an I-γ-cover. If Un : n ∈ ω and Vn : n ∈ ω are shrinkable I-γ-covers, then Un ∩ Vn : n ∈ ω is a shrinkable I-γ-cover. Finally, if Un : n ∈ ω is an I-γ-cover and Un ⊆ Vn for each n, then Vn : n ∈ ω is an I-γ-cover.

• For two families A, B of sequences of subsets of X, we

introduce similarly as M. Scheepers did, the property S1(A,B) as follows: for every sequence Un,m : m ∈ ω : n ∈ ω of sequences from A, there exists a sequence mn : n ∈ ω of natural numbers such that Un,mn : n ∈ ω ∈ B. If a topological space X possesses the property S1(A,B) we shall say that X is an S1(A,B) -space.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

• A topological space X (or a subset with the subspace

topology) with the property S1(Ω, Γ) is a γ-space, where Ω is the family of all ω-covers 4 of X. The basic results concerning the existence of γ-spaces were proved by F . Galvin and A.W. Miller.

4An open cover A of X is an ω cover, if for every finite F ⊆ X, there exists

a set U ∈ A such that F ⊆ U.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

• A topological space X (or a subset with the subspace

topology) with the property S1(Ω, Γ) is a γ-space, where Ω is the family of all ω-covers 4 of X. The basic results concerning the existence of γ-spaces were proved by F . Galvin and A.W. Miller.

• As above, one can easily show that if X is

an S1(I-Γ,J -Γ)-space, then for every sequence Un,m : m ∈ ω : n ∈ ω of sequences of I-γ-covers there exists a sequence mn : n ∈ ω of natural numbers such that mn

J

− → ∞ and Un,mn : n ∈ ω is a J -γ-cover.

4An open cover A of X is an ω cover, if for every finite F ⊆ X, there exists

a set U ∈ A such that F ⊆ U.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 6

Bukovski and hales had found a characterization of wQN-spaces by covers, namely wQN ≡ S1(Γsh, Γ). We can show similar result.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

Theorem 6

Bukovski and hales had found a characterization of wQN-spaces by covers, namely wQN ≡ S1(Γsh, Γ). We can show similar result. Theorem If X is a normal topological space, then the following are equivalent: (i) X is an (I, sJ )wQN-space, (ii) X is an S1(I-Γsh, J -Γ)-space.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

References [1] M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007), 715–729. [2] N. Bourbaki, TOPOLOGIE G´

EN´ ERALE, Hermann & Sci, Paris

1940. [3] Z. Bukovsk´ a, Quasinormal convergence, Math. Slovaca 41 (1991), 137–146. [4] L. Bukovsk´ y, THE STRUCTURE OF THE REAL LINE, Monog. Mat., vol 71, Springer-Birkh¨ auser, Basel, 2011. [5] L. Bukovsk´ y, I. Recław, M. Repick´ y, Spaces not distinguishing pointwise and quasi-normal convergence of real

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

functions, Topology Appl. 41 (1991), 25–40. [6] L. Bukovsk´ y, J. Haleˇ s, QN-spaces, wQN-spaces and covering properties, Topology Appl. 154 (2007), 848–858. [7] H. Cartan, Th´ eorie des filtres, C. R. Acad. Sci. Paris 205 (1937), 595–598. [8] H. Cartan, Filtres et ultrafiltres, C. R. Acad. Sci. Paris 205 (1937), 777–779. [9] ´

• A. Cs´

asz´ ar, M. Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungarica 10 (1975), 463–472. [10] P . Das, Certain types of open covers and selection principles using ideals, Houston J. Math. 39 (2013), 637–650. [11] P . Das, D. Chandra, Spaces not distinguishing pointwise

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

and I-quasinormal convergence of real functions, Comment

• Math. Univ. Carolin. 54 (2013), 83–96.

[12] P . Das, D. Chandra, (I, J )-quasinormal spaces, manuscript, 2013. [13] G. Di Maio, Lj.D.R. Koˇ cinac, Statistical convergence in topology, Topology Appl. 156 (2008), 28–45. [14] R. Filip´

• w, M. Staniszewski, On ideal equal convergence,
• Cent. Eur. J. Math. 12 (2014), 896–910.

[15] F . Galvin, A.W. Miller, γ-sets and other singular sets of real numbers, Topology Appl. 17 (1984), 145–155. [16] J. Jasinski, I. Recław, On spaces with the ideal convergence property, Colloq. Math. 111 (2008), 43–50.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

[17] P . Kostyrko, T. ˇ Sal´ at, W. Wilczy´ nski, I-convergence, Real

• Anal. Exchange 26 (2000/2001), 669–685.

[18] F . Rothberger, Sur les familles ind´ enombrables de suites de nombress naturels et les probl´ emes concernant la propri´ et´ e C, Proc. Cambridge Phil. Soc. 37 (1941), 109–126. [19] M. Sakai, The sequence selection properties of Cp(X), Topology Appl. 154 (2007), 552–560. [20] M. Scheepers, Gaps in ωω, In: Set theory of the Reals (

• H. Judah, ed.), Israel Mathematical Conference Proceedings 6

(1993), 439–561. [21] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69 (1996), 31–62.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

[22] M. Scheepers, Sequential convergence in Cp(X) and a covering property, East-West J. Math. 1 (1999), 207–214. [23] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74. [24] J. ˇ Supina, Ideal QN-spaces, preprint.

Ideal quasi-normal convergence and related notions

Basic notions Properties of the Ideal and Decompositions Equivalences with classical notions (I,J )QN, (I,J )wQN and properties of Cp(X) I-γ-covers

THANK YOU FOR YOUR ATTENTION

Ideal quasi-normal convergence and related notions