For a topological space $X$ a topological contraction on $X$ is a closed mapping $f:X\to X$ such that for every open cover of $X$ there is a positive integer $n$ such that the image of the space $X$ via the $n$th iteration of $f$ is a subset of some element of the cover. Every topological contraction in a compact $T_1$ space has a unique fixed point. As in the case of metric spaces and the classical Banach fixed point theorem, this analogue of Banach's theorem is true not only in compact but also in complete (here in the sense of \v{C}ech) $T_1$ spaces. We introduce a notion of weak topological contraction and in Hausdorff spaces we prove the existence of a unique fixed point for such continuous and closed mappings without assuming completeness or compactness of the space considered. These theorems are applied to prove existence of fixed points for mappings on compact subsets of linear spaces with weak topologies and for compact monoids. We also prove some fixed point results for $T_1$ locally Hausdorff spaces and, introduced here, peripherally Hausdorff spaces. An iterated function system on a topological space, IFS, is a finite family of closed mappings from the space into itself. It is contractive if for every open cover of $X$ for some positive integer $n$ the image of $X$ via a composition of $n$ mappings from the IFS is contained in an element of the cover. We show that in $T_1$ compact topological spaces the Hutchinson operator of a contractive IFS may not be closed as the mapping in the hyperspace of closed subsets of the space. Nevertheless, the Hutchinson operator of a contractive IFS has always a unique fixed point.

This work is motivated by the two classical theorems on inscribing rectangles and squares into large subsets of the plane: Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem, we prove that for every Borel subset of the plane with uncountably many vertical sections that are positive (with respect to measure or category) contains a rectangle $$P\times B$$ P × B where P is perfect and B is Borel and positive. We also present a variant of Eggleston Theorem regarding the $$\sigma -$$ σ - ideal $$\mathcal {E}$$ E generated by closed sets of measure zero. Furthermore, we show that every comeager (resp. conull) subset of the plane contains a rectangle $$[T]\times H$$ [ T ] × H , where T is a splitting tree containing a Silver tree and H is comeager (resp. conull). Additionally, we establish a joint generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $$[T]\times H$$ [ T ] × H modulo diagonal, where T is a uniformly perfect tree, H is comeager (resp. conull) and $$[T]\subseteq H$$ [ T ] ⊆ H .

We work in the Baire space ℤ^ω equipped with the coordinate-wise addition +. Consider a σ-ideal ℐ and a family 𝕋 of some kind of perfect trees. We are interested in results of the form: for every A∈ℐ and a tree T∈𝕋 there exists T'∈𝕋, T'⊆ T such that A+[T']+[T']+… +[T']_n–times∈ℐ for each n∈ω. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and ω-Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard σ-ideal ℳ of meager sets, we also analyze ℳ_- and fake null sets 𝒩. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cicho\'n, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski--Burstin representable ideals, as well as to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.

In this paper we investigate the action of Polish groups (not necessary abelian) on an uncountable Polish spaces. We consider two main situations. First, when the orbits given by group action are small and the second when the family of orbits are at most countable. We have found some subgroups which are not measurable with respect to a given σ-ideals on the group and the action on some subsets gives a completely nonmeasurable sets with respect to some σ-ideals with a Borel base on the Polish space. In most cases the general results are consistent with ZFC theory and are strictly connected with cardinal coefficients. We give some suitable examples, namely the subgroup of isometries of the Cantor space where the orbits are sufficiently small. In the opposite case we give an example of the group of the homeomorphisms of a Polish space in which there is a large orbit and we have found the subgroup without Baire property and a subset of the mentioned space such that the action of this subgroup on this set is completely nonmeasurable set with respect to the σ-ideal of the subsets of first category.

In $T_1$ compact topological spaces the Hutchinson operator of a contractive IFS (iterated function system; a finite family of closed mappings from the space into itself) may not be closed. Nevertheless, the Hutchinson operator of a contractive IFS has always a unique fixed point.

We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.

A σ-ideal I on a Polish group (X,+) has Smital Property if for every dense set D and a Borel I-positive set B the algebraic sum D + B is a complement of a set from I. We consider several variants of this property and study their connections with countable chain condition, maximality and how well they are preserved via Fubini products.

Abstract We prove that if there exists a continuous surjection from a metric compact space 𝑋 onto a product X × T X\times T , where 𝑇 is a T 1 T_{1} second countable topological space which has the cardinality of the continuum, then there exists a surjection from 𝑋 onto the product X × [ 0 , 1 ] X\times[0,1] , where the interval [ 0 , 1 ] [0,1] is equipped with the usual Euclidean topology.

We prove that if there exists a continuous surjection from a metric compact space X onto a product X x T, where T is a T-1 second countable topological space which has the cardinality of the continuum, then there exists a surjection from X onto the product X x [0, 1], where the interval [0, 1] is equipped with the usual Euclidean topology.

We construct a metric continuum X such that there exists a continuous surjection F from X onto X with the inverse image of every point of size of the continuum and such that for no nontrivial topological space T there exists a surjection from X onto X×T. In particular, F cannot be a composition of a surjection from X onto X×T and the projection onto X, where T is a topological space of size of the continuum with at least one nonempty open set not equal to the whole space. This is a sharpening of the result obtained in [2].

Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup \Delta$. We consider generalizations of this theorem by replacing a perfect square with a rectangle $A\times B$, where $A$ and $B$ are bodies of other types of trees with $A\subseteq B$. In particular, we show that for every comeager $G_\delta$ set $G\subseteq \omega^\omega\times \omega^\omega$ there exist a Miller tree $M$ and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq G\cup\Delta$ and that $P$ cannot be a Miller tree. In the case of measure we show that for every subset $F$ of $2^{\omega}\times 2^\omega$ of full measure there exists a uniformly perfect tree $P\subseteq 2^{<\omega}$ such that $[P]\times[P]\subseteq F\cup\Delta$ and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions of the real line from which we derive nonstandard proofs of Mycielski-like theorems via Shoenfield Absoluteness Theorem.

Abstract In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$ , $m_0$ , $l_0$ , $cl_0$ , $h_0,$ and $ch_0$ . We show that there exists a subset of the Baire space $\omega ^\omega ,$ which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of ${\mathbb {T}}$ -Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees ${\mathbb {T}}$ . We also obtain a result on ${\mathcal {I}}$ -Luzin sets, namely, we prove that if ${\mathfrak {c}}$ is a regular cardinal, then the algebraic sum (considered on the real line ${\mathbb {R}}$ ) of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$ , $l_0,$ and $cl_0$ .

Abstract We examine images of Bernstein sets via continuous mappings. Among other results, we prove that there exists a continuous function f : ℝ → ℝ {f\colon\mathbb{R}\to\mathbb{R}} that maps every Bernstein subset of ℝ {\mathbb{R}} onto the whole real line. This gives the positive answer to a question of Osipov.

A subset of an abelian group is midpoint-free if it contains no three distinct elements a, b, c such that a + b = 2c. We study midpoint-free sets in various classical topological groups. For every infinite cardinal kappa <= c, we show that the real line can be partitioned into K-many maximal midpoint-free sets. Examples of closed maximal midpoint-free subsets are given for topological groups such as R, C, S-1, S-1 x S-1. Finally, among sets that are not regular, such as nonmeasurable sets, Bernstein sets, and Luzin sets, we study instances which are midpoint-free.

ABSTRACT. In this note we consider a Marczewski like nonmeasurable sets (with respect to trees) which forms m.a. $d$ . family in Baire space. Here we show that under assumption that $\omega_{1}=b$ there is a m.a. $d$ . family in Baire space which is not $s$-measurable (here we can replace $s$-nonmeasurable by -nonmeasurable or mnonmeasurable). Moreover it is relatively consistent with ZFC theory that $\omega_{1}<\mathfrak{d}\leq \mathfrak{c}$ and there is m.a. $d$ . family in Baire space which is not measurable with respect to family of all complete Laver trees in $\omega^{\omega}.$

Let X be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel sigma-additive measure mu which is good in the sense that for any clopen subsets U, V subset of X with mu(U) < mu(V) there is a clopen set W subset of V with mu(W) = mu(U). We study sigma-ideals with Borel base on X which are invariant under the action of the group H-mu(X) of measure-preserving homeomorphisms of (X, mu), and show that any such sigma-ideal I is equal to one of seven s-ideals: {sic}, [X](<=omega), epsilon, M boolean AND N, M, N, or [X](<= c). Here [X](<=kappa) is the ideal consisting of subsets of cardinality <=kappa in X, M is the ideal of meager subsets of X, N - {A subset of X : mu(A) - 0} is the ideal of null subsets of (X, mu), and epsilon is the sigma-ideal generated by closed null subsets of (X, mu).

We study and classify topologically invariant σ-ideals with Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal cardinalities of the families of Cantor sets covering the unit interval and the Hilbert cube are the same.

We study and classify topologically invariant sigma-ideals with an analytic base on Euclidean spaces, and evaluate the cardinal characteristics of such ideals.

In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by Miller trees. With the mentioned ideals one can consider s, l and m-measurablility. We have shown that there exists a subset A of the Baire space which is s, l and m nonmeasurable at the same time. Moreover, A forms m.a.d. family which is also dominating. We show some examples of subsets of the Baire space which are measurable in one sense and nonmeasurable in the other meaning. We also examine terms nonmeasurable and completely nonmeasurable (with respect to several ideals with Borel base). There are several papers about finding (completely) nonmeasurable sets which are the union of some family of small sets. In this paper we want to focus on the following problem: "Let P be a family of small sets. Is it possible that for all A which is a subset of P, union of A is nonmeasurable implies that union of A is completely nonmeasurable?" We will consider situations when P is a partition of R, P is point-finite family and P is point-countable family. We give an equivalent statement to CH using terms nonmeasurable and completely nonmeasurable.