Artur Bartoszewicz

Strong algebrability of sets of sequences and functions

We introduce a notion of strong algebrability of subsets of linear algebras. Our main results are the following. The set of all sequences from c0 which are not summable with any power is densely strongly c–algebrable. The set of all sequences in l∞ whose sets of limit points are homeomorphic to the Cantor set is comeager and strongly calgebrable. The set of all non-measurable functions from RR is strongly 2–algebrable. These results complete several ones by other authors, within the modern context of lineability.

Large free linear algebras of real and complex functions

Let X be a set of cardinality κ such that κω=κ. We prove that the linear algebra RX (or CX) contains a free linear algebra with 2κ generators. Using this, we prove several algebrability results for spaces CC and RR. In particular, we show that the set of all perfectly everywhere surjective functions f:C→C is strongly 2c-algebrable. We also show that the set of all functions f:R→R whose sets of continuity points equals some fixed Gδ set G is strongly 2c-algebrable if and only if R⧹G is c-dense in itself.

Multigeometric sequences and Cantorvals

For a sequence x ∈ l1\c00, one can consider the achievement set E(x) of all subsums of series Σn=1∞x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie J.A., Nymann J.E., The topological structure of the set of subsums of an infinite series, Colloq. Math., 1988, 55(2), 323–327] we describe families of sequences which contain, according to our knowledge, all known examples of x with E(x) being Cantorvals.

Additivity and lineability in vector spaces

Abstract Gamez-Merino, Munoz-Fernandez and Seoane-Sepulveda proved that if additivity A ( F ) > c , then F is A ( F ) -lineable where F ⊆ R R . They asked if A ( F ) > c can be weakened. We answer this question in negative. Moreover, we introduce and study the notions of homogeneous lineability number and lineability number of subsets of linear spaces.

Large Function Algebras with Certain Topological Properties

Let be a family of continuous functions defined on a compact interval. We give a sufficient condition so that contains a dense -generated free algebra; in other words, is densely -strongly algebrable. As an application we obtain dense -strong algebrability of families of nowhere Holder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras within where or . We prove that the set of perfectly everywhere surjective functions together with the zero function contains a -generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra of with generators which is closed in the pointwise topology and, for any function in this algebra, there is an open set such that is a Bernstein set.

On similarity between topologies

Let T1 and T2 be topologies defined on the same set X and let us say that (X, T1) and (X, T2) are similar if the families of sets which have nonempty interior with respect to T1 and T2 coincide. The aim of the paper is to study how similar topologies are related with each other.

On generating regular Cantorvals connected with geometric Cantor sets

Abstract We show that the Cantorvals connected with the geometric Cantor sets are not achievement sets of any series. However many of them are attractors of IFS consisting of affine functions.

Large free sets in powers of universal algebras

We prove that for each universal algebra

Nonseparable spaceability and strong algebrability of sets of continuous singular functions

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