Covering an uncountable square by countably many continuous functions

We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpi\'nski's theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.Comment: Added new results (9 pages

Large separated sets of unit vectors in Banach spaces of continuous functions

The paper concerns the problem whether a nonseparable \C(K) space must contain a set of unit vectors whose cardinality equals to the density of \C(K) such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of \C(K) spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly 2.Comment: The second version does not contain new results, but it is reorganized in order to distinguish our main contributions from what was essentially know

Classification of one dimensional dynamical systems by countable structures

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation

Topological fractals revisited

We prove that every Peano continuum with uncountably many local cut points is a topological fractal. This extends some recent results and it partially answers a conjecture by Hata. We also discuss the number of mappings which are necessary for witnessing the structure of a topological fractal

Homogenita topologických struktur

In the present work we study those compacti cations such that every autohomeomorphism of the base space can be continuously extended over the compacti cation. These are called H-compacti cations. We characterize them by several equivalent conditions and we prove that H-compacti cations of a given space form a complete upper semilattice which is a complete lattice when the given space is supposed to be locally compact. Next, we describe all H-compacti cations of discrete spaces as well as of countable locally compact spaces. It is shown that the only H-compacti cations of Euclidean spaces of dimension at least two are one-point compacti cation and the Cech-Stone compacti cation. Further we get that there are exactly 11 H-compacti cations of a countable sum of Euclidean spaces of dimension at least two and that there are exactly 26 H-compacti cations of a countable sum of real lines. These are all described and a Hasse diagram of a lattice they form is given.Department of Mathematical AnalysisKatedra matematické analýzyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Non-cut, shore and non-block points in continua

In a nondegenerate continuum we study the set of non-cut points. We show that it can be stratified by inclusion into six natural subsets (containing also non-block and shore points). Among other results we show that every nondegenerate continuum contains at least two non-block points. Our investigation is further focused on both the classes of arc-like and circle-like continua