Countable Ordinal Number Research Articles

On metric spaces with given transfinite asymptotic dimensions

For every countable ordinal number ξ, we construct a metric space Xξ whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both ξ.

On linear continuous operators between distinguished spaces $$C_p(X)$$

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space $$C_{p}(X)$$ is distinguished if and only if X is a $$\Delta $$ -space. If there exists a linear continuous surjective mapping $$T:C_p(X) \rightarrow C_p(Y)$$ and $$C_p(X)$$ is distinguished, then $$C_p(Y)$$ also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator $$T:C_p(X) \rightarrow C_p(Y)$$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $$C_p([1,\alpha ])$$ , where $$\alpha $$ is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping $$T:C_p([1,\alpha ]) \rightarrow C_p(Y)$$ is given. We also observe that for every countable ordinal $$\alpha $$ all closed linear subspaces of $$C_p([1,\alpha ])$$ are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of $$\Delta $$ -spaces we prove that a linear continuous surjection $$T:C_p(X) \rightarrow C_k(X)_w$$ , where $$C_k(X)_w$$ denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact $$X \subset {\mathbb {R}}^n$$ ).

Some properties related to the Cantor-Bendixson derivative on a Polish space

We show a necessary and sufficient condition for any ordinal number to be a Polish space. We also prove that for each countable Polish space, there exists a countable ordinal number that is an upper bound for the first component of the Cantor-Bendixson characteristic of every compact countable subset of the aforementioned space. In addition, for any uncountable Polish space, for every countable ordinal number and for each nonzero natural number, we show the existence of a compact countable subset of this space such that its Cantor-Bendixson characteristic equals the previous pair of numbers. Finally, for every Polish space, we determine the cardinality of the partition, up to homeomorphisms, of the set of all compact countable subsets of the aforesaid space.

The relationship between asymptotic decomposition properties

In this paper, we study the relationship between complementary-finite asymptotic dimension and transfinite asymptotic dimension. We prove that for every metric space X, coasdim(X)≤γ implies trasdim(X)≤γ for every countable ordinal number γ. Therefore, the metric space with complementary-finite asymptotic dimension has asymptotic property C. Furthermore, we construct an example of a metric space satisfying asymptotic property C and finite decomposition complexity with arbitrary high asymptotic dimension growth, which shows that asymptotic property C and finite decomposition complexity need not imply sub-exponential dimension growth in general.

Levels of Generalized Expansiveness

We study a class of generalized expansive dynamical systems for which at most countable orbits can be accompanied by an arbitrary given orbit. Examples of different levels of generalized expansiveness are constructed. When the dynamical system considered has countable cardinality, several characterizations of generalized expansive homeomorphism are provided, and as a consequence examples of dynamical systems with van der Waerden depth equal to any given countable ordinal number are demonstrated, which solves open questions existing in the literature.

α-Large families and applications to Banach space theory

The notion of α-large families of finite subsets of an infinite set is defined for every countable ordinal number α, extending the known notion of large families. The definition of the α-large families is based on the transfinite hierarchy of the Schreier families Sα, α<ω1. We prove the existence of such families on the cardinal number 2ℵ0 and we study their properties. As an application, based on those families we construct a reflexive space X2ℵ0α, α<ω1 with density the continuum, such that every bounded non-norm convergent sequence {xk}k has a subsequence generating ℓ1α as a spreading model.

Q α-Normal Families and Entire Functions

For every countable ordinal number α, we construct an entire function f = fα, such that the family \({\lbrace f(nz):n\in N\rbrace}\) is exactly Qα-normal in the unit disk.

On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trind X ⩽ α . W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trInd X ⩽ λ . T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dim W X ⩽ α where dim W is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dim C of the covering dimension. As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dim C X ⩽ α belonging to some absolute Borel class.

A Degree of Nonlocal Connectedness

To any continuum X we assign an ordinal numher (or the symbol (0) s(X), called the degree of nonlocal connectedness of X. We show that (1) the degree cannot be increased under continuous surjections; (2) for hereditarily unicoherent continua X, the degree of a subcontinuum of X is less than or equal to s(X); (3) s(C(X)) :::; s(X), where C(X) denotes the hyperspace of subcontinua of a continuum X. We also investigate the degrees of Cartesian products and inverse limits. As an application we construct an uncountable family of metric continua X homeomorphic to C(X). Introduction. The idea of using ordinal numbers as a measure of some local or global properties of (compact) spaces is not new. Usually these properties are related to (non-)connectedness, and the defined measure can be used as a tool in studying various other properties of investigated spaces, both structural (internal) and mapping (external) ones. For example, Iliadis in (14) defines the notion of a normal sequence for hereditarily decomposable and hereditarily unicoherent metric continua (i.e., for >.-dendroids) as follows. Let X be such a continuum. A continuum HeX is said to be in I(X) if, given any decomposition of X into finitely many subcontinua, H is contained in one element of the decomposition. Let I: = {Her: a .} be a transfinite sequence of subcontinua of X, where >. is some countable ordinal number. Then I: is called a nonnal sequence if (i) Ho = X, (ii) H{3 = I(Her) for ordinals (3 = a + 1, (iii) H{3 = n{Her : a . the continuum Her is nondegenerate. The least upper bound (or minimum) k(X) of the lengths of all normal sequences in X is called the depth of X. The concept was used to study various phenomena in >'-dendroids. For its modification, see (31).

The depth of centres of maps on dendrites

AbstractXiong proved that if f: I → I is any map of the unit interval I, then the depth of the centre of f is at most 2, and Ye proved that for any map f: T → T of a finite tree T, the depth of the centre of f is at most 3. It is natural to ask whether the result can be dendrites. In this note, we show that there is dendrite D such that for any countable ordinal number λ there is a map f: D →D such that the depth of centre of f is λ. As a corollary, we show that for any countable ordinal number λ there is a map (respectively a homeomorphism) f of a 2-dimensional ball B2 (respectively a 3-dimensional ball B3) such that the depth of centre of f is λ.

On σ-porous sets and Borel sets

Even if σ-porous sets are neglected, for each countable ordinal number α there exist Borel sets which are not of the class α.

Ordinal rankings on measures annihilating thin sets

We assign a countable ordinal number to each probability measure which annihilates all H H -sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman’s conjecture. Similar results are obtained for measures annihilating Dirichlet sets.

The set of continuous functions with everywhere convergent Fourier series

This paper deals with the descriptive set theoretic properties of the class EC of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in C(T). A natural coanalytic rank function on EC is studied that assigns to each ƒ Є EC a countable ordinal number, which measures the complexity of the convergence of the Fourier series of ƒ. It is shown that there exist functions in EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with ω_1 distinct levels.

Algebraic p-adic expansions

We describe a theory p-adic expansions for algebraic p-adic numbers in which countable ordinal number index sets and p-adically nonconvergent expansions are needed. This theory is used to compute parts of the expansions of the p”th roots of unity and to answer a question of Koblitz [l] about transcendence degrees. We shall use the following notation. Let Cl!, be the p-adic completion of Q for a prime p, a field of cardinality 2 ‘O. Let Qpp be the algebraic closure of Q,, and let CP be its p-adic completion, which is an algebraically closed complete field with a valuation uniquely extended from QP and is also of cardinality 2% Let D, = {z E C,: 1 z 1 < 1 }. Let Qyram be the field consisting of the set of x E% such that the ring extension Q, AD, c Q,(x) n D, is unramified. Let Gym be the p-adic completion of Qym, considered as a subfield of CP. Let 1 *In denote any primitive n th root of unity in C,. We choose in CP one of each rational power of p which together compose a multiplicative group p”, and consider these choices to be fixed throughout this paper. We introduce the field of p-adic ordinal series (or “padic Malcev-Neumann series”) which are formal sums 1 cipr,, where the ii’s are roots of unity in CP of order prime to p (i.e., among representatives for ST), and { ri} is a well-ordered subordered-set of Q. We let an ordinal number i index the term whose preceding terms are ordered by i. If x = x cjp’, then we let xi = cj<, ijp’ denote an initial sum. One checks by translinite induction that the field operations analogous to those of QP are well defined and that this field is complete non-archimedean valued. -. We now define (using the axiom of choice) an embedding of Q, mto the field of p-adic ordinal series which extends the inclusion of pQ and the obvious embedding of Qp, given by p-adic expansion. We successively extend the embedding through a chain of finite field extensions; we continue the argument only for the first extension Q, c QJx). Let f(t) be the minimal polynomial of x over Q,. To determine the p-adic expansion of x,