March 29, 2018Sebastian Lindner,Faculty of Mathematics and Computer ScienceUniwersity of Lodz

Self-presentation

Contents

1 Diplomas and degrees 1

2 Employment 2

3 Scientific achievement forming the basis for habilitation pro-cedure 2

4 Detailed description of the achievement 34.1 H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2 H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.3 H3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.4 H4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.5 H5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.6 H6 i H7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Description of the other published works 12

1 Diplomas and degrees

• Philosophy Doctor in Mathematics - obtained in 2003 upon the thesisentitled:,,O pewnych wasnosciach miar indukowanych przez wahanie”Supervisor: Prof. Dr.hab. Jacek Hejduk

• Master of Science - obtained in 1998 upon the tesis entitled ,,Wasnoscipewnego sigma-idea lu podzbiorow prostej.”Supervisor: Prof. Dr. hab. W ladys law Wiliczynski.

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2 Employment

Since May 1998 I have been working in the Chair of Real Functions of theFaculty of Mathematics, first as an assistant, since 2004 as a lecturer, thensince 2016 as senior lecturer.

3 Scientific achievement forming the basis for

habilitation procedure

Scientific achievement forming the basis for a habilitation procedure is thecollection of papers entitled:

On relations between topology, measurable space and algebraicalstructures

containing the following articles:

H1 S. Lindner, M. Terepeta On the position of abstract density topologiesin the lattice of all topologies FILOMAT 30:2(2016) 281–286.

H2 S. Lindner, M. Terepeta Almost Semi-Correspondence Georgian Math-ematical Journal 24:3 (2016), 439–446.

H3 S. Lindner Resolvability properties of similar topologies Bulletin of theAustralian Mathematical Society 92:3 (2015), 470–477.

H4 G. Horbaczewska, S. Lindner Resolvability of measurable spaces Bul-letin of the Australian Mathematical Society 94:1 (2016), 70–79.

H5 A. Karasinska, S. Lindner On special saturated sets, Bulletin dea Societedes Sciences et des Lettres de d. Srie: Recherches sur les Dformations66 (3) (2016), 79–86

H6 G. Horbaczewska, S. Lindner Density, Smital Property and Quasicon-tinuity Bulletin of the Australian Mathematical Society 97:2 (2018),246–252.

H7 G. Horbaczewska, S. Lindner, On sets which can be moved away fromsets of a certain family, Journal of Mathematical Analysis and Appli-cations, Vol. 472, Issue 1, (2019) 231–237.

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4 Detailed description of the achievement

Many interesting properties can be observed, if a space is equipped with twodifferent structures connected in a certain way. The classical example is anotion of a topological group, i.e. a space equipped with the group structureand a topology, such that group operations are continuous. Many beautifulltheorems encourage to investigate this area. Let me mention here:

• The Steinhausa theorem, which says that if a set A ⊂ R has a positiveLebesgue measure, then the set A−A contains some neighbourhood of0,

• The Smitala lemma which says that for A,B ⊂ R, if A is of a positiveLebesgue measure and B is dense, then A+B is of a full measure,

• Galvin, Mycielski and Solovay theorem, that characterizes sets of strongmeasure zero on the real line as those, which can be translated out ofan arbitrary first cathegory set.

The papers discussed below fall within this area.

4.1 H1

In the paper H1 a connection between the notion of abstract density topol-ogy (ADT) and the relation of semi-correspondence between topologies isestablished.

The notion of ADT was introduced by Haupt i Pauc in 1952 ([18]). Letus recall the definitions in a current terminology:

Let A be a field of subsets of a set X and let I ⊂ A be a proper ideal.For A,B ⊂ X let the statment A ∼ B denote that A4B ∈ I. An operatorΦ: A → 2X is called a lower density operator , if for any A,B ∈ A thefollowing conditions are satisfied:

(a) Φ(∅) = ∅, Φ(X) = X;

(b) Φ(A ∩B) = Φ(A) ∩ Φ(B);

(c) A ∼ B ⇒ Φ(A) = Φ(B);

(d) A ∼ Φ(A) (the analog of the Lebesgue’a density theorem).

Let me recall that by measurable hull of a set A ⊂ X we denote sucha set B ∈ A, that A ⊂ B and for any C ∈ A the condition C ⊂ B \ A

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implies C ∈ I. A space (X,A, I) has the hull property if every its subset hasa measurable hull.

If a space (X,A, I) has the hull property and Φ is a lower density operatoron that space, then the family T = {A ∈ A : A ⊂ Φ(A)} forms a topology.Each topology which can be expressed this way is called an abstract densitytopology (ADT). ADT were examined between others in papers [13], [11],[14], [19].

From the other side let us consider an arbitrary topological space (X, τ).A set A ⊂ X is called semi-open if A ⊂ Cl(Int(A)). Consider the fam-ily of all topologies on the space X. We say that topologies τ1 and τ2 aresemi-correspondent (τ1 ∼sc τ2) if their families of semi open sets coincide.Equivalence classes of this relation where examined in papers [10], [25], [24]i [17]. In particular in [25] and [10] the authors proved independently, thateach eqivalence class of ∼sc has its greatest element.

The main achievement of the paper H1 is a characterization of semi opensets in abstract density topologies (theorem 2.6):

Theorem Let Φ be a lower density operator on the space (X,A, I). Then

A ∈ SO(X,T Φ) ⇐⇒ A ∈ A ∧ A ∩ Φ(A′) = ∅.

This result lets us observe (in proposition 2.7) that if two ADTs, definedon the same space (X,A, I) are semi-correspondent, then they are equall toeach other.

As a consequence we obtained the most important result of the pa-per (corollary 3.4) that in each equivalence class of the relation of semi-correspondence there is exactly one ADT, and it is the greatest (with respectto inclusion) element of the class. This result joins two ideas developedindependently in topology since ’60.

4.2 H2

The main inspiration of the note H2 was to correct a mistake, found in thepaper of T. R. Hamletta [17].

In the paper H1 two realtions between topologies were considered: therelation ∼sc discussed above and the relation of simmilarity: Two topologiesτ1 and τ2 are called simmilar (τ1 ∼s τ2), if their families of sets with nonemptyinterior coincide. This relation was also examined by many authors, betweenothers in papers [4] and [28].

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In general the condition (τ1 ∼sc τ2) is stronger than (τ1 ∼s τ2). In hispaper T. R. Hamlett has formulated two theorems (no 3 i 4), which thesissaid that certain two topologies are in the relation ∼sc . In our paper wegave counter-examples and then we proved weaker theorems (theorems 3.5 i3.6). To achieve that we introduced the new relation, weaker than ∼sc butstronger than ∼s .

The rest of the paper is devoted to some examples of topologies remain-ing in discussed relations. Topologies generated by various lower densityoperators were considered.

4.3 H3

The notion of resolvability of a topological space was introduced by Hewittin 1943 ([20]). Since then spaces with very good resolvability (maximallyresolvable, extra-resolvable) were deeply investigated. Also badly resolvablespaces, i.e. OHI (open hereditary irresolvable) and HI (hereditary irresolv-able) were examined (see [1], [2], [8], [16], [29], [30] ).

Recall that a topological space (X,T ) is κ-resolvable, if there exists afamily of cardinality of κ of parwise disjoint dense sets. It is easy to see thatκ has to be less than or equall to the smallest cardinality of a nonemptyopen set, denoted by ∆(X,T ). A dense in itself ∆(X,T )-resolvable space iscalled maximally resolvable. A space is extra-resolvable, if there is a familyof cardinality greater than ∆(X,T ), of dense sets which are pairwise almostdisjoint, i.e. the intersection of two sets is always nowhere dense.

In the paper H3 I characterized resolvable spaces and OHI spaces us-ing notions refering to the pair (NB,ND) where NB is the field of sets ornowhere dense boundary and ND is the ideal of nowhere dense sets (theo-rems 3.2 i 3.3): Let H(A) denote the ideal of sets belonging hereditary toA – it is the greatest ideal contained in A. A topological space is resolvableif and only if H(NB) = ND. A space is strongly irresolvable if and only ifNB = 2X .

This result enabled me to observe that the property OHI is invariant withrespect to the relation of similarity (described in the previous section), whilethe property HI and the property of submaximality are not (corollary 3.4and example 3.6).

In the second part of the paper I examined which pair (A, I) can beexpressed in the form (A, I) = (NB(τ),ND(τ)) for some topology τ. Wesay that such pairs are topological.

In order to achieve this a notion of the ideal, which is small in a field,was defined:

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Definition An ideal I is small in a field A, if for every set from the familyA \H(A) one can find the subset belonging to the family H(A) \ I.

In property 4.4. it is shown that if A 6= 2X , and an ideal I is small inA, then the pair (A, I) is not topological. In virtue of that proposition weobtain that the pair (L, Iα) composed of the field of Lebesgue measurablesets and the ideal of sets having Hausdorff dimension less than or equall toα is not topological (corollary 4.6).

4.4 H4

In the paper H3 resolvability properties of topological space were expressedusing some properties of the pair (NB,ND) composed of some field andsome ideal of subsets of the space. In the sequel a space (X,A, I) where Ais a field of subsets of X and I ⊂ A is an ideal, is called a measurable space.

In the paper H4 we examine questions concerning resolvability of mea-surable spaces not necessarily being topological. These problems are strictlyconnected with the operators S i S0 introduced by Burstin and Marczewski([7], [34]).

The idea of resolvability of families which are not topologies is not new –it was introduced in the paper of Jimenez and Malykhin [22].

Each topological space can be uniquely expressed as a sum X = F ∪G, where F is a closed resolvable subspace and G – hereditary irresolvablesubspace. Such pair of sets is called Hewitt decomposition of a topologicalspace. In our paper we define the notion of Hewitts decomposition of ameasurable space – however the decomposition is not unique. We prove thecondition sufficient for existence of such decomposition (theorem 3.2): if Ais σ-field, I – σ-ideal, and the space (X,A, I) fulfills the countable chaincondition, then it has Hewitt decomposition.

Then a method of construction of spaces not having the Hewitt decom-position is shown (theorem 3.6).

We introduce a notion of weak resolvability of the measurable space,which, under the assumption of existence of Hewitt decomposition, is equiv-alent to resolvability:

Definition A space (A, I) is weakly resolvable, if it does not contain astrongly irresolvable set from A \ I.

We prove that the necessary condition of weak resolvability is H(A) = I.The results of this part of the paper are gathered on the following diagram.

The left column presents properties of the measurable space (A, I). The rightone – respective properties of the space (S(A\I), S0(A\I)) generated with

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use of operators S i S0 of Marczewski and Burstin:

S(F) := {E ⊂ X : ∀A∈F ∃B∈F (B ⊂ A ∩ E ∨ B ⊂ A \ E)}

andS0(F) := {E ⊂ X : ∀A∈F ∃B∈F B ⊂ A \ E}.

H(A) denotes as previously the ideal of sets hereditarily belonging to A. Thecondition expressed as (A, I) – inner means that the space (A, I) is innerMB representable, i.e. there exists a family F ⊂ A of nonempty subsets ofX such that S(F) = A and S0(F) = I.

The diagram should be read as follows: from the fact that (A, I) is weaklyresolvable it follows that H(A) = I, and the inverse implication holds underthe additional condition of inner MB representability of the space (A, I).

H(A) = I H(S(A \ I)) = S0(A \ I)

(A, I) - inner.

(A, I) - weakly res. (S(A \ I), S0(A \ I)) - weakly res.

(A, I) - Hewitt

(A, I) - resolvable (S(A \ I), S0(A \ I)) - resolvable

?

In the last part of the paper strongly irresolvable spaces were character-ized under the assumption of inner MB representability (it is the analog ofElkin‘s result for topological spaces).

4.5 H5

In 1902 H. Lebesgue defined his measure, in 1905 Vitali constructed a non-measurable set, and in 1908 E. Van Vleck gave the example of saturatednon-measurable set, i.e. a set A such that both A and it’s complement haveinner measure equall to 0 ([36]).

Let D denote the set of dyadic rationals. The set E ⊂ R constructed byVan Vleck has the following properties:

(I) it is invariant with respect to dyadic rational translations,

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(II) sets E, −E and D form a decomposition of the space R.

In the paper H5 similar sets in abelian semi-topological groups equippedwith the structure of a measurable space invariant with respect to the groupoperation were examined.

In the first part of the article we give a sufficient condition for a space sothat those two properties imply the saturated non-measurability of the setE :

Corollary Let a space (X,A, I) be invariant and D be a dense subgroupof X such thar D ∈ I. If (X,A, I) has the Stainhaus property, then each sethaving the properties (I) and (II) is (X,A, I)-saturated non-measurable.

In the second part of the paper we concentrate on a wider class of sets– so called Archimedean sets. A subset A of a semi-topological group X iscalled Archimedean if there exists a dense set D, such that A + d = A foreach d ∈ D. We suggested a weaker wersion of the Smital property:

Definition A space (X,A, I) has the property (wS), if for each A ∈ A \ Iand each dense subgroup D the condition (A+D)′ ∈ I holds.

Theorem The following statements are equivalent:

1. (X,A, I) has the property (wS);

2. each Archimedean set A fullfills exactly one of the following conditions:

(a) A ∈ I.

(b) X \ A ∈ I.

(c) A is (X,A, I)-saturated non-measurable.

At the end of the paper we give interesting example of a space, thatfullfills (wS), but it does not satisfy the general Smital property. In order toconstruct it we used a tool supplied by Cichon and Szczepaniak in [9].

4.6 H6 i H7

The last two papers are devoted to operators, which transform families ofsubsets of given subspace X. Considered operators are similar to each other:each of them is of the form O : 22X → 22X , and each has the following prop-erties:

Let F ,G be arbitrary nonempty families of nonempty subsets of X. Then

• G ⊂ O(F) ⇐⇒ F ⊂ O(G),

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• G ⊂ F =⇒ O(F) ⊂ O(G),

• O(3)(F) = O(F).

The last condition means that iterating the operator we will eventually obtaintwo families of sets such that F = O(G) and G = O(F).

In the first part of H6 the following operator is examined:Let X be a nonempty set, F ⊂ 2X \ {∅}, F 6= ∅. We say that P ⊂ X is

dense with respect to the family F , if it intersects with any member of thefamily. By D(F) we shall denote the family of all sets dense with respect toF .

In the paper basic properties of the operator D were examined. In par-ticular it was proved, that for any family F of nonempty subsets of X thefollowing condition holds:

S0(D(F)) = S0(F).

(S0 is again the Burstin-Marczewski operator). This result has been alreadynoticed in the paper of A. Nowik [26], but we menaged to prove it with noadditional asumptions.

A notion of a set dense with respect to some family is of course strictlyconnected with the idea of resolvability, discussed in the previous papers.One of the beautiful results in this field is the theorem of A. Illanes from1996 r. ([21]): If a topological space is n-resolvable for any integer n – thenit is also ℵ0-resolvable. In our paper we constructed a quite simple exampleto show that the Illanes result cannot be generalized to any family of sets.Recently the Illanes result has ben generalized by D. Fremlin (preprint canbe found on https://www1.essex.ac.uk/maths/people/fremlin/n17910.pdf).

A starting point for the second part of the paper was already mentionedSmital lemma:

Lemma For arbitrary set A ⊂ R of a positive Lebesgue measure and anydense set D the set A + D is of full measure (i.e. it’s complement is a nullset).

Observe that the above property connects three structures on the realline: the topology, the algebraic structure and the structure of measurablespace. Our objective was to discuss analogous properties in an arbitrarycommutative group and for arbitrary field and ideal of subsets of the group.In order to do this we introduced an operator DS :

Let (X,+) be a commutative group and let I ⊂ P (X) be an arbitraryideal of subsets of X. For any family F ⊂ P (X) \ {∅} let

DSI(F) = {D ⊂ X : ∀F∈F(F +D)′ ∈ I}.

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Such operator also has properties defined at the beginning of this section.In the sequel two more notations will be needed: Let F be an arbitrary

family of subsets of a group X. Let us denote F = {F1 − F2 : F1, F2 ∈ F},and let F∗ = F \ {∅}.

In my opinion the most significant result of the second part of H6 is acharacterization of some intetesting class of semitopological groups using theoperator DS.

Recall that a group equipped with a topology is called semi-topological ifthe topology is invariant with resect to group operations and it is called topo-logical, if the group operations are continuous with respect to the topology.Each topological group is semitopological. Let X, Y be topological spaces. Amap f : X → Y is called quasicontinuous in x ∈ X, if for any neighbourhodU of x and any neighbourhood V of f(x) there exists a nonempty open setG ⊂ U such that f(G) ⊂ V.

We extracted the class of semitopological groups in which addition isquasicontinuous. The following theorem gives a characterization of thosegroups (theorem 4.1):

Theorem Let (X,+, τ) be a semitopological group. Let ND denote theideal of nowhere dense sets. Then the following statements are equivalent:

(1) the operation + : X ×X → X is quasicontinuous,

(2) the family τ∗ is coinitial to the family τ∗,

(3) DSND(τ∗) = D(τ∗).

In the paper a non-trivial example of such group was supplied.Let L,N denote the σ−field of Lebesgue measurable sets and σ−ideal of

Lebesgue null sets, respectively. The Smital lemma says thatDN (L \ N ) ⊃ D,where D denotes the family of dense subsets of R. It is easy to observe thatthe last inclusion can be replaced by equality. From Smital lemma it followsalso that the family DN (D) contains all sets of a positive inner measure. Butthis is not a characterization: in the paper [12] an example of a set of innermeasure zero belonging to DN (D) has been supplied.

The paper H6 ends with an open problem of characterization of the familyD

(2)N (L \ N ) = DN (D).

100 years ago E. Borel published the paper [6], in which he introduceddefinition of strong measure zero set. Let us recall that a subset A of ametric space is of strong measure zero, if for any sequence εn of positivenumbers there exists the sequence of balls K(xn, rn) such that rn ≤ εn andA ⊂

⋃nK(xn, rn). At the same paper Borel made the hypotesis that the

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only strong measure zero sets in R are countable sets. The family of strongmeasure zero sets will be denoted by SMZ.

In 1928 W. Sierpinski proved under the assumption of continuum hypote-sis the existence of uncountable SMZ sets, and in 1976 Laver constructedthe model of set theory, where Borel hypotesis is satisfied.

Well known theorem of F. Galvina, J. Mycielski and R. Solovay ([15])characterizes strong measure zero sets on the real line in the following way:

SMZ = {A ⊂ R : ∀M∈KA+M 6= R},

where K is the family of first cathegory sets. The elegance of this resulthas prompted many authors to examine families that can be defined in thesimilar way.

Among others in the paper of J. Pawlikowski and M. Sabok [27] theoperator ∗ : 22X → 22X defined in the following way:

F∗ = {A ⊂ R : ∀F∈FA+ F 6= R}

has been considered. According to this definition the GMS theorem says thatSMZ = K∗.

In the first part of tje paper H7 properties of the operator ∗ in com-mutative groups and commutative topological groups were examined. Inparticular the families Fin∗ and Count∗ were characterized, where Fin andCount stand for the families of finite and countable sets, respectively.

In the second part of the article we restricted our attention to the realline. The key result of the paper is corollary 10:

Theorem Assume continuum hypotesis. Let I ⊂ 2R be a proper σ-idealinfariant with respct to translations and reflections satisfying the conditioncof(I) ≤ c. Then I∗∗ = I.

The main idea of the proof has been found in the paper of T. Weiss [35].This result enabled us to observe that under CH the family of first cathe-

gory sets K can be characterized as SMZ∗. Similarily N = SM∗, whereN and SM denote families of Lebesgue null sets and strong first cathegorysets, respectively.

Let us consider now the family of meager-additive sets: a set A ⊂ R ismeager additive if A+K ∈ K for every K ∈ K. This condition is a strongerversion of the tesis of GMS theorem. The family of meager-additive sets formsa σ−ideal contained in SMZ. This family has been also intensively studied– among others S. Shelah in the paper [31] proved (under CH) the existenceof uncountable subset of the Cantor space, which is meager-additive. Lastyear the characterizations of both families were given by O. Zindulka [37].

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The corollary 10 let us (under CH) characterize the family of meager-additive sets in a new way (theorem 14):

Let O denote a family of subsets of R defined in the following way:O = {A ⊂ R : ∃K∈K∃S∈SMZA ⊂ K + S.} In virtue of the GMS theoremthe family O forms a propper σ-ideal.

Theorem The following conditions are equivalent:

(1) P ∈ O∗ ,

(2) P is SMZ−additive,

and the condition

(3) P is meager-additive

implies (1) and (2). Under CH all three conditions are equivalent. Theanalogous result has been established for the family of null-additive sets.

5 Description of the other published works

A1 J. Hejduk, S. Lindner, On the Hashimoto topology with respect to anextension of the Lebesgue measure, Tatra Mt. Math. Publ. 24 (2002),147-151

A2 S. Lindner, Topologies of Hashimoto type with respect to sigma-ideal ofcountable sets, Folia Mathematica 11(2004) 55-58

Let τ be a topology and I be a σ−ideal on X. If the family τI ={G \ I : G ∈ τ, I ∈ I} forms a topology, it is called a Hashimototopolgy. In the paper A1 topologies generated by a semi-lower den-sity operator with respect to an arbitrary complete extension of theLebesgue measure were characterized (theorem 9). In the paper A2three Hashimoto topologies were defined and examined.

B1 S. Lindner, Generalization of the Banach Indicatrix Theorem, RealAnal. Exchange 27 (2001), no. 2, 721–724.

B2 S. Lindner, Additional properties of the measure vf , Tatra Mt. Math.Publ. 28(2004), 199–205

The Banach indicatrix theorem connects the total variation of the con-tinuous function g on the interval and the Lebesgue integral of theindicatrix of g.. In the paper B1 the measure vf connected with thetotal variation of the continuous function f has been defined. The main

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result (theorem 1) allows to express the total variation of the composi-tion f ◦g as the integral of the indicatrix of the function g with respectto the measure vf . In the next paper connections between propertiesof the function f and the measure vf were established.

C1 J. Jachymski, M. Lindner, S. Lindner On Cauchy type characterizationsof continuity and Baire one functions Real Analysis Exchange Volume30, Number 1 (2004-2005), 339-346.

C2 M. Lindner, S. Lindner Characterizations of Some Subclasses of theFirst Class of Baire Real Anal. Exchange Vol. 36, No 2 (2010), 499-506.

In the paper [23] Lee, Tang and Zhao gave very elegant characterizationof functions of the first Baire class:

Let X and Y be complete and separable metric spaces, thenthe function f : X → Y is in the first Baire class if and onlyif:

for any ε > 0 there exists a positive function δ defined on Xsuch that for arbitrary x1, x2 ∈ X the condition dX(x1, x2) <min{δ(x1), δ(x2)} implies that dY (f(x1), f(x2)) < ε.

In the papers C1 and C2 this idea has been examined. Other operatorsinstead of ,,min” were discussed. We were also trying to characterizesubclasses of the first Baire class which can be obtained by imposingadditional conditions on the function δ.

D1 S. Lindner, W. Wilczynski, On points of the regular density TatraMountains Mathematical Publications, 52 (2012), 917.

D2 S. Lindner, The regular density on the plane, Annales UniversitatisPaedagogicae Cracoviesis Studia Mathematica X (2011), 79-87.

D3 S. Lindner, Remark on the regular density on the plane, MonografiaReal functions, density topology and related topics (2011), Lodz, WydawnictwoUniwersytetu Lodzkiego.

In the above sequence of papers the following modification of the def-inition of a density point is considered: Let A ⊂ R be a Lebesguemeasurable set, x0 ∈ R. Consider fx0 : [0, 1]→ [0, 1] defined as follows:

fx0(h) =µ(A ∩ [x0 − h, x0 + h])

2h

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for h > 0 i fx0(0) = 1. The set A has an ordinary density point at x0

if and only if the function fx0 is continuous at 0. The point x0 will becalled a point of a regular density, if additionally the function fx0 hasbounded variation on [0, 1].

In the paper D1 it was proved that a regular density points of a setcoincide with its O’Malley points. In the next paper it was provedthat such characterization does not hold if subsets of the space Rn areconsidered for n > 1. Moreover it turned out that the regular densityoperator has quite ,,bad” properties in this case – in particular a regulardensity point of a set A ⊂ R2 does not have to be a regular densitypoint of a superset of A.

E1 S. Lindner, M. Terepeta Algebrability within the class of Baire 1 func-tions Lithuanian Mathematical Journal, 55 (2015), 393–401, doi: 10.1007/s10986-015-9287

In the paper problems of algebrability of some classes of functions,contained in the family of the first Baire class, were considered.

Definition Let A be a subset of a commutative algebra. Then A isκ-algebrable, if A∪{0} contains a κ-geneated algebra B, i.e. a minimalsystem of generators of B is of the cardinality κ. In the paper [5] A.Bartoszewicz and S. G la,b introduced a notion of strong algebrability.A set A is strongly κ-algebrable if A∪{0} contains κ-generated algebraB isomorfic with free algebra.

A problem of algebrability is usually considered with respect to classesof obiects of ,,bad” properties, for example functions of the first Baireclass but not continuous. Considered sets do not form, in general,an algebraical structure, in particular – they do not contain the zeroelement.

A very good tool used in proving of algebrability of some class of func-tions is a lemma supplied by M. Balcerzak, A. Bartoszewicz and M.Filipczak in the paper [3]:

Definition We say that f : R→ R is exponential-like of the range m,if it can be expressed as f(x) =

∑mi=1 aie

βix for some different non-zeronumbers β1, . . . , βm and some non-zero a1, . . . , am.

Theorem [Proposition 7, [3]] Let us consider a family F ⊂ R[0,1], andsuppose that there exists a function F ∈ F such that f ◦F ∈ F \{0} foran arbitrary exponential-like function f : R → R. Then F is stronglyc-algebrable.

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Let Cd, Cnat, B1 and DB1 denote class of functions continuous withrespect to the density topology (i.e. approximately continuous), con-tinuous with respect to the natural topology, of the first class of Baireand of the first class of Baire with Darboux property, respectively.

In the paper E1 we have used the tool mentioned above to prove that

– The family Cd \ Cnat contains the lower semi-lattice (with respectto inclusion) consisting of the continuum many classes of func-tions, and each nonempty difference of two classes is strongly c-algebrable (section 2.1);

– The family DB1 \ Cd contains the lower semi-lattice (with respectto inclusion) consisting of the continuum many classes of func-tions, and each nonempty difference of two classes is strongly c-algebrable (section 2.2);

– The class B1 \ DB1 contains the family of classes of functions,lineary ordered to type ω1, and any nonempty difference of twoclasses is strongly c-algebrable (section 3).

To prove the first two facts, more universal theorem was used (theorem2). It gives the way to prove the strong c-algebrbility of a difference oftwo class of functions continuous with respect to topologies generatedby two different lower density operators. This theorem was later usedby R. Wiertelak and F. Strobin in they paper [33].

F1 J. Hejduk, S. Lindner, A. Loranty, On lower density type operators andtopologies generated by them, Filomat 32:14 (2018), 49494957

Let X be a nonempty set. Let I be an arbitrary ideal of subsets of X.Then Sm = {A ⊂ X : A ∈ I ∨ X \ A ∈ I} is the smallest field ofsubsets of X containing I. Let S be an arbitrary field of subsets of Xsuch that Sm ⊂ S. The symbol A ∼ B denotes that A4B ∈ I. Let usconsider Φ : S → P (X) and the following set of conditions:

a. Φ(∅) = ∅, Φ(X) = X,

b. Φ(A ∩B) = Φ(A) ∩ Φ(B),

c. A ∼ B =⇒ Φ(A) = Φ(B),

d. Φ(A) \ A ∈ I,d*. Φ(A) ∼ A,

e. Φ(A ∪B) = Φ(A) ∪ Φ(B).

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Each operator satisfying conditions a – c is called semi lower densityoperator – SLDO. An operator satisfying conditions a – d is calledalmost lower density operator – ALDO. An operator ALDO, whichsatisfies the condition (d*) is called a lower density operator (LDO).Finally – an operator fullfiling all conditions is called a lifting.

In the paper F1 we consider the natural order in the space of operators.We prove that maximal operator ALDO is a liftingiem. We chatacterizespaces, in which the greatest operator LDO exists.

At the same time we prove theorems concerning the existence of thegreatest (or maximal) topology generated by those operators.

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