Zero-dimensional Research Articles

Compact spaces homeomorphic to their respective squares

We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of Włodzimierz J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.

A compact space K is Corson compact if and only if Cp(K) has a dense lc-scattered subspace

We establish that a compact space K is Corson compact if and only if it can be embedded in a topological group which is a W-space in the sense of Gruenhage. As a application of this result we show that K is Corson compact if and only if it embeds in Cp(X) for a Lindelöf scattered space X. We also prove that a zero-dimensional compact space K is Corson if and only if Cp(K,{0,1}) is a continuous image of a Lindelöf scattered space. Besides, a zero-dimensional countably compact space X is ω-monolithic if and only if Cp(X) has a dense functionally countable subspace.

Uniqueness and homogeneity of non-separable Urysohn universal ultrametric spaces

We first introduce the new notion of petaloid ultrametric spaces, which is intended to be a standard class of non-separable Urysohn universal ultrametric spaces. Next we prove that all petaloid spaces are isometric to each other and homogeneous for all finite subspaces (and compact subspaces). Moreover, we show that the following spaces are petaloid, and hence they are isometric to each other and homogeneous: (1) The space of all continuous functions, whose images contain the zero, from the Cantor set into the space of non-negative real numbers equipped with the nearly discrete topology, (2) the space of all continuous ultrametrics on a zero-dimensional infinite compact metrizable space, (3) the non-Archimedean Gromov–Hausdorff space, and (4) the space of all maps from the set of non-negative real numbers into the set of natural numbers whose supports are finite or decreasing sequences convergent to the zero.

Two topologies on the set of minimal prime elements of a continuous frame

In this paper, we discuss the properties of the hull-kernel topology and the inverse topology on the set [Formula: see text] of minimal prime elements of a continuous frame [Formula: see text]. We prove that the set [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, is a [Formula: see text]-space. Moreover, we obtain that there is a bijective correspondence between the set [Formula: see text] of all maximal Scott open filters of [Formula: see text] and the set [Formula: see text] when [Formula: see text] is a stably continuous frame. By considering the bijective correspondence between the sets [Formula: see text] and [Formula: see text], we propose some sufficient conditions for the topological spaces [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, to be sober, Hausdorff, compact, extremely disconnected and zero-dimensional spaces, respectively, when [Formula: see text] is a stably continuous frame.

On the largest regular subring of C(X)

We call a real-valued function f on a topological space X a locally constant if each point x ∈ X has a neighborhood on which f is constant. The set of all locally constant functions f: X → ℝ is denoted by E(X). We observe that E(X) is the largest von Neumann regular subring of C(X) containing ℝ. We show that for every space X, there is a zero-dimensional space Y such that E(X) ≅ E(Y) and E(X) determines the topology of X if and only if X is a zero-dimensional space. By an example, it turns out that E(X) is not always of the form C(Y) for some space Y, but whenever X is a locally connected space, we observe that E(X) is a C(Y), where Y is a discrete space. Finally some von Neumann local subrings of C(X) containing ℝ are introduced and topological spaces X are characterized for which these subrings coincide with the smallest one. Note that a ring R is said to be a von Neumann local ring if for each a ∈ R either a or 1 − a has von Neumann inverse.

Plasticity of the unit ball of some C(K) spaces

We show that if K is a compact metrizable space with finitely many accumulation points, then the closed unit ball of C(K) is a plastic metric space, which means that any non-expansive bijection from BC(K) onto itself is in fact an isometry. We also show that if K is a zero-dimensional compact Hausdorff space with a dense set of isolated points, then any non-expansive homeomorphism of BC(K) is an isometry.

Zero-divisor graph and comaximal graph of rings of continuous functions with countable range

In this paper, two outwardly different graphs, namely, the zero-divisor graph [Formula: see text] and the comaximal graph [Formula: see text] of the ring [Formula: see text] of all real-valued continuous functions having countable range, defined on any zero-dimensional space [Formula: see text], are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness are concerned. However, the study reveals that the zero-divisor graph [Formula: see text] of an intermediate ring [Formula: see text] of [Formula: see text] is complemented if and only if the space of all minimal prime ideals of [Formula: see text] is compact. Moreover, [Formula: see text] is complemented when and only when its subgraph [Formula: see text] is complemented. On the other hand, the comaximal graph of [Formula: see text] is complemented if and only if the comaximal graph of its over-ring [Formula: see text] is complemented and the latter graph is known to be complemented if and only if [Formula: see text] is a [Formula: see text]-space. Indeed, for a large class of spaces (i.e. for perfectly normal, strongly zero-dimensional spaces which are not P-spaces), [Formula: see text] and [Formula: see text] are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilized to establish that for a discrete space [Formula: see text], [Formula: see text] (= [Formula: see text]) and [Formula: see text] (= [Formula: see text]) are isomorphic, if [Formula: see text] is at most countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.

Some observations on a clopen version of the Rothberger property

In this paper, we prove that a clopen version $S_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ of the Rothberger property and Borel strong measure zeroness are independent. For a zero-dimensional metric space $(X,d)$, $X$ satisfies $S_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ if, and only if, $X$ has Borel strong measure zero with respect to each metric which has the same topology as $d$ has. In a zero-dimensional space, the game $G_1(\mathcal{O}, \mathcal{O})$ is equivalent to the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ and the point-open game is equivalent to the point-clopen game. Using reflections, we obtain that the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ is undetermined.

Zero-dimensional σ-homogeneous spaces

All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. Inspired by this theorem, we obtain the following results:•Assuming AD, every zero-dimensional space is σ-homogeneous,•Assuming AC, there exists a zero-dimensional space that is not σ-homogeneous,•Assuming V=L, there exists a coanalytic zero-dimensional space that is not σ-homogeneous. Along the way, we introduce two notions of hereditary rigidity, and give alternative proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is σ-homogeneous.

Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the Lovász Local Lemma (the LLL for short), which was introduced by Erdős and Lovász in the mid-1970s. Here we develop a version of the LLL that can be used to prove the existence of continuous colorings. We then give several applications in Borel and topological dynamics.•Seward and Tucker-Drob showed that every free Borel action Γ↷X of a countable group Γ admits an equivariant Borel map π:X→Y to a free subshift Y⊂2Γ. We give a new simple proof of this result.•We show that for a countable group Γ, Free(2Γ) is weakly contained, in the sense of Elek, in every free continuous action of Γ on a zero-dimensional Polish space. This fact is analogous to the theorem of Abért and Weiss for probability measure-preserving actions and has a number of consequences in continuous combinatorics. In particular, we deduce that a coloring problem admits a continuous solution on Free(2Γ) if and only if it can be solved on finite subgraphs of the Cayley graph of Γ by an efficient deterministic distributed algorithm (this fact was also proved independently and using different methods by Seward). This establishes a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing.

On comeager sets of metrics whose ranges are disconnected

For a metrizable space X, we denote by Met(X) the space of all metric that generate the same topology of X. The space Met(X) is equipped with the supremum distance. In this paper, for every strongly zero-dimensional metrizable space X, we prove that the set of all metrics whose ranges are closed totally disconnected subsets of the line is a dense Gδ subspace in Met(X). As its application, we show that some sets of universal metrics are meager in spaces of metrics.

Evolutionary Approach to Biological Homochirality

We study a very simple linear evolutionary model based on distribution of protocells by total enantiomeric excess and without any mutual inhibition and show that such model can produce two species with values of total enantiomeric excess in each of the species approaching pm 1 when there is a global Lleftrightarrow D symmetry. We then consider a scenario when there is a small external global asymmetry factor, like weak interaction, and show that only one of the species remains in such a case, and that is the one, which is more efficient in replication. We perform an estimate of the time necessary to reach homochirality in such a model and show that reasonable assumptions lead to an estimate of around 300 thousand years plus or minus a couple of orders of magnitude. Despite this seemingly large time to reach homochirality, the model is immune to racemization because amino acids in the model follow the lifespan of the protocells rather than the time needed to reach homochirality. We show that not needing mutual inhibition in such evolutionary model is due to the difference in the topology of the spaces in which considered model and many known models of biological homochirality operate. Bifurcation-based models operate in disconnected zero-dimensional space (the space is just two points with enantiomeric excess equal -1 and 1), whereas considered evolutionary model (in its continuous representation) operates in one-dimensional connected space, that is the whole interval between -1 and 1 of total enantiomeric excess. We then proceed with the analysis of the replication process in non-homochiral environment and show that replication errors (the probability to attach an amino acid of wrong chirality) result in a smooth decrease of replication time when total enantiomeric excess of the replicated structure moves away from zero. We show that this decrease in replication time is sufficient for considered model to work.

Regularity of certain commutative Banach rings

Let (A,‖⋅‖) be a commutative unital Banach ring. Let M(A) be the Berkovich spectrum of A, i.e., the set of all bounded multiplicative non-zero semi-norms defined on A which is endowed with the pointwise convergence topology. As in the case of complex Banach algebras, we introduce the notions of the hull-kernel topology on M(A) and the regularity of a commutative unital Banach ring. We show that the Banach algebra of all bounded continuous k-valued functions defined on a zero-dimensional topological space is regular, where k is a complete valuation field. Hence, we see that the Berkovich spectrum of this algebra is a Hausdorff space in the hull-kernel topology. Next, we use the tool of Gelfand transform to characterize a Banach ring of finite Berkovich spectrum. From this, we see that if A is uniform, i.e., ‖f2‖=‖f‖2 for all f in A, and has finite Berkovich spectrum, then it is regular.

On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)

In this article we consider some relations between the topological properties of the spaces X and Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of Min(Cc (X)) is equivalent to the von-Neumann regularity of qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of Cc(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if 𝑋 is an Fc-space, then Min(Cc (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with β0X. Finally, by introducing zoc-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected.

Entropy pair realization

AbstractWe show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.

Closed ideals in the functionally countable subalgebra of C(X)

In this paper, closed ideals in Cc(X), the functionally countable subalgebra of C(X), with the mc-topology, is studied. We show that ifX is CUC-space, then C*c(X) with the uniform norm-topology is a Banach algebra. Closed ideals in Cc(X) as a modified countable analogue of closed ideals in C(X) with the m-topology are characterized. For a zero-dimensional space X, we show that a proper ideal in Cc(X) is closed if and only if it is an intersection of maximal ideals of Cc(X). It is also shown that every ideal in Cc(X) with the mc-topology is closed if and only if X is a P-space if and only if every ideal in C(X) with the m-topology is closed. Moreover, for a strongly zero-dimensional space X, it is proved that a properly closed ideal in C*c(X) is an intersection of maximal ideals of C*c(X) if and only if X is pseudo compact. Finally, we show that if X is a P-space and F is an ec-filter on X, then F is an ec-ultrafilter if and only if it is a zc-ultrafilter.

Abundance of independent sequences in compact spaces and Boolean algebras

It follows from a theorem of Rosenthal that a compact space is ccc if and only if its every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces {mathcal {C}} we define its orthogonal {mathcal {C}}^perp as the class of all compact spaces for which every continuous image in {mathcal {C}} is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and ccc compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where it is abundant. We study these problems for zero-dimensional compact spaces with the aid of Boolean algebras, and show the main difficulties that arise when we deal with general settings. Our main results are the constructions of several relevant examples.

Topological Transcendental Fields

This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2ℵ0 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 22ℵ0 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic.

ON THE m_c-TOPOLOGY ON THE FUNCTIONALLY COUNTABLE SUBALGEBRA OF C(X)

In this paper, we consider the $m_c$-topology on $C_c(X)$, the functionally countable subalgebra of $C(X)$. We show that a Tychonoff space $X$ is countably pseudocompact if and only if the $m_c$-topology and the $u_c$-topology on $C_c(X)$ coincide. It is shown that whenever $X$ is a zero-dimensional space, then $C_c(X)$ is first countable if and only if $C(X)$ with the $m$-topology is first countable. Also, the set of all zero-divisors of $C_c(X)$ is closed if and only if $X$ is an almost $P$-space. We show that if $X$ is a strongly zero-dimensional space and $U$ is the set of all units of $C_c(X)$, then the maximal ring of quotients of $C_c(U)$ and $C_c(C_c(X))$ are isomorphic.

Some observations on the mildly Menger property and topological games

In this paper, we defined two new games - the mildly Menger game and the compact-clopen game. In a zero-dimensional space, the Menger game is equivalent to the mildly Menger game and the compact-open game is equivalent to the compact-clopen game. An example is given for a space on which the mildly Menger game is undetermined. Also we introduced a new game namely K-quasi-component-clopen game and proved that this game is equivalent to the compact-clopen game. Then we proved that if a topological space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy in the mildly Menger game.