Product Topology Research Articles

Combinatorial properties on nodec countable spaces with analytic topology

We study some variations of the product topology on families of clopen subsets of 2N×N in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle q+.

On Completeness of Sliced Spaces under the Alexandrov Topology

We show that in a sliced spacetime ( V , g ) , global hyperbolicity in V is equivalent to T A -completeness of a slice, if and only if the product topology T P , on V, is equivalent to T A , where T A denotes the usual spacetime Alexandrov “interval” topology.

Li-Yorke Chaos for ultragraph shift spaces

Recently, in connection with C*-algebra theory, the first author and Danilo Royer introduced ultragraph shift spaces. In this paper we define a family of metrics for the topology in such spaces, and use these metrics to study the existence of chaos in the shift. In particular we characterize all ultragraph shift spaces that have Li-Yorke chaos (an uncountable scrambled set), and prove that such property implies the existence of a perfect and scrambled set in the ultragraph shift space. Furthermore, this scrambled set can be chosen compact, which is not the case for a labelled edge shift (with the product topology) of an infinite graph.

The Power of Sectoral Geographical Centrality in Global Production

Power is a key component in understanding and analyzing global production and the governance structures of Global Value Chains. In this paper, we propose a novel analytical link between the power dynamics in GVCs and the network configuration of their respective production topology. Our proposed link is based on the notion of positional power according to which power is associated with the centrality of a sector with regards to the production process, the sector belongs to. Using global input-output data, we show that the network structure of global production is associated with the global distribution of profits among national economic sectors and, consequently, influences the power relations and thus the governance structures of supply networks. More specifically, we find a high correlation between the distribution of profits and a sector’s position in global production, captured by its total strength centrality. Based on this, we provide a quantitative measure of positional power within global production and its governance structures.

Marginals and the product strong lifting problem

We study the permanence of strong liftings for products of topological probability spaces. We get results to the positive for the 'usual product' or for the product σ-algebra obtained from it by adjoining the two-sided nil-null sets under assumptions implying that the product topology is contained in this product σ-algebras. For the construction of strong liftings in the product the application of marginals is basic as well as that of the fundamental properties of lifting topologies.

Ultragraph shift spaces and chaos

Motivated by C*-algebra theory, ultragraph edge shift spaces generalize shifts of finite type to the infinite alphabet case. In this paper we study several notions of chaos for ultragraph shift spaces. More specifically, we show that Li-Yorke, Devaney and distributional chaos are equivalent conditions for ultragraph shift spaces, and characterize this condition in terms of a combinatorial property of the underlying ultragraph. Furthermore, we prove that such properties imply the existence of a compact, perfect set which is distributionally scrambled of type 1 in the ultragraph shift space (a result that is not known for a labelled edge shift (with the product topology) of an infinite graph).

Selectors for dense subsets of function spaces

Let USCp⋆(X) be the topological space of real upper semicontinuous bounded functions defined on X with the subspace topology of the product topology on RX. Φ˜↑,Ψ˜↑ are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of USCp⋆(X), respectively. We prove several equivalent assertions to that USCp⋆(X) satisfies the selection principles S1(Φ˜↑,Ψ˜↑), including a condition on the topological space X.We prove similar results for the topological space Cp⋆(X) of continuous bounded functions.Similar results hold true for the selection principles Sfin(Φ˜↑,Ψ˜↑).

Topologies on quotient space of matrices via semi‐tensor product

AbstractAn equivalence of matrices via semi‐tensor product (STP) is proposed. Using this equivalence, the quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices lead to the product topology and quotient topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the quotient space. This metric leads to a metric topology. A comparison for these three topologies is presented. Some topological properties are revealed.

Continuous Implementation with Simple Mechanisms

We revisit the idea that robustness concerns can provide a foundation for simple mechanisms. Formally, we consider the robust implementation notion of continuous implementation (Oury and Tercieux, 2012), where continuity is with respect to players’ hierarchy of beliefs. We define simplicity as (the truthful equilibrium of) a direct revelation mechanism. We show that these two desiderata are at odds. For continuity with respect to a finer topology on the hierarchy of beliefs (which is a weaker requirement), the uniform-weak topology, we show that there is a revelation principle. Conversely, for continuity with respect to the coarser product topology (a stronger requirement), we show that no such revelation principle holds. At a technical level, our characterization of continuous implementation in the product topology uses a novel contagion argument which does not require the existence of “crazy types” as is commonly assumed, and may be of independent interest.

Topology-Selective Ullmann Coupling on Metal Surfaces by Precursor Design and Adsorbate-Substrate Interaction: Towards the Control over Polymer versus Macrocycle Formation.

In polymerization reactions, controlling the formation of open-chain versus cyclic product topologies is necessary because of the different properties of polymer chains and macrocycles. Here, we report a topology-selective Ullmann coupling on metal surfaces with control of the ring/chain competition. The precursor employed is 4,4''-dibromo-ortho-terphenyl (DBOTP), which features a 60° bent feature and polymerizes into zigzag polyphenylene chains on both Au(111) and Ag(111) surface via Ullmann coupling. However, the cyclotrimerization of the precursor occurs only on Ag(111) but not Au(111). It is proposed that the cyclotrimerization reaction on Au(111) is blocked, because the necessary C-C coupling of two carbon radicals with different vertical heights is unfavored. Such height difference stems from the intrinsic steric repulsion between the two ortho-substituted phenyl groups. On Ag(111), the stronger adsorbate-substrate interaction reduces the extent of the tilting of the phenyl group, resulting in a smaller height difference of the carbon radicals and consequently in the increased probability for the formation of the cyclic trimer.

Product of generalized quasi-uniform spaces

In this paper, we introduce product \(g\)-quasi uniformity and show that product \(g\)-quasi uniformity induces the generalized product topology. We also provide a necessary and sucient condition for a mapping into product \(g\)-quasi uniform space to be \(g\)-quasi uniformly continuous. Further, we establish the equivalence between completeness of product \(g\)-quasi uniform space and that of the component spaces.

Josefson\u2013Nissenzweig property for C_{p}-spaces

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or ell _{2}. The aim of the paper is to study a natural variant of this result for the space C_{p}(X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C_{p}(X)-spaces. We prove: for a Tychonoff space X the space C_p(X) satisfies the JNP if and only if C_p(X) has a quotient isomorphic to c_{0}:={(x_n)_{nin mathbb N}in mathbb {R}^mathbb {N}:x_nrightarrow 0} (with the product topology of mathbb {R}^mathbb {N}) if and only if C_{p}(X) contains a complemented subspace isomorphic to c_0. The last statement provides a C_{p}-version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space c_{0} with the sup-norm topology. For a pseudocompact space X the space C_p(X) has the JNP if and only if C_p(X) has a complemented metrizable infinite-dimensional subspace. An example of a compact space K without infinite convergent sequences with C_{p}(K) containing a complemented subspace isomorphic to c_{0} is given.

Data centres as logistical facilities: Singapore and the emergence of production topologies

Data centres mobilise server-client architectures to disperse and draw in labour from across industries and nations. In doing so, they provide an infrastructural fix for capitalist actors seeking to bypass traditional labour actions, by designing logistical routes around which to redirect production processes. In this article, we build on research that investigates the data centre industry in Singapore to consider how these facilities drive processes of global circulation and establish new kinds of labour relations and processes. We point to limits in conceptualising these relations according to dominant models of the supply chain or the production network. We argue that understanding the client footprint enabled by data centres as a form of territory allows us to approach these facilities as political institutions that influence the operations of power across wide geographical vistas.

Evaluation of Large-Scale Production of Chitosan Microbeads Modified with Nanoparticles Based on Exergy Analysis

Novel technologies for bio-adsorbent production are being evaluated on the lab-scale in order to find the most adequate processing alternative under technical parameters. However, the poor energy efficiency of promising technologies can be a drawback for large-scale production of these bio-adsorbents. In this work, exergy analysis was used as a computer-aided tool to evaluate from the energy point of view, the behavior of three bio-adsorbent production topologies at large scale for obtaining chitosan microbeads modified with magnetic and photocatalytic nanoparticles. The routes were modeled using an industrial process simulation software, based on experimental results and information reported in literature. Mass, energy and exergy balances were performed for each alternative, physical and chemical exergies of streams and chemical species were calculated according to the thermodynamic properties of biomass components and operating conditions of stages. Exergy efficiencies, total process irreversibilities, energy consumption, and exergy destruction were calculated for all routes. Route 2 presents the highest process irreversibilities and route 3 has the highest exergy of utilities. Exergy efficiencies were similar for all simulated cases, which did not allow to choose the best alternative under energy viewpoint. Exergy sinks for each topology were detected. As values of exergy efficiency were under 3%, it was shown that there are process improvement opportunities in product drying stages and washing water recovery for the three routes.

Results on a Pre-T_2 Space and Pre-Stability

This paper contains an equivalent statements of a pre- space, where are considered subsets of with the product topology. An equivalence relation between the preclosed set and a pre- space, and a relation between a pre- space and the preclosed set with some conditions on a function are found. In addition, we have proved that the graph of is preclosed in if is a pre- space, where the equivalence relation on is open.
 On the other hand, we introduce the definition of a pre-stable ( pre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that a pre-stable ( pre-stable) set is positively invariant (invariant), and we add a condition on this set to prove the converse. Finally, a relationship between, (i) a pre-stable ( pre-stable) set and its component (ii) a pre- space and a (positively critical point) critical point, are gotten.

Results on a Pre-T_2 Space and Pre-Stability

This paper contains an equivalent statements of a pre- space, where are considered subsets of with the product topology. An equivalence relation between the preclosed set and a pre- space, and a relation between a pre- space and the preclosed set with some conditions on a function are found. In addition, we have proved that the graph of is preclosed in if is a pre- space, where the equivalence relation on is open.
 On the other hand, we introduce the definition of a pre-stable ( pre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that a pre-stable ( pre-stable) set is positively invariant (invariant), and we add a condition on this set to prove the converse. Finally, a relationship between, (i) a pre-stable ( pre-stable) set and its component (ii) a pre- space and a (positively critical point) critical point, are gotten.

An example of a capacity for which all positive Borel sets are thick

On the Cantor cube {0,1}N with the standard product topology we construct a finite Choquet capacity with respect to the family of all compact sets such that every compact set of positive capacity contains continuum many pairwise disjoint compact subsets of positive capacity.

On Soft Rough Topology with Multi-Attribute Group Decision Making

Rough set approaches encounter uncertainty by means of boundary regions instead of membership values. In this paper, we develop the topological structure on soft rough set ( SR -set) by using pairwise SR -approximations. We define SR -open set, SR -closed sets, SR -closure, SR -interior, SR -neighborhood, SR -bases, product topology on SR -sets, continuous mapping, and compactness in soft rough topological space ( SRTS ). The developments of the theory on SR -set and SR -topology exhibit not only an important theoretical value but also represent significant applications of SR -sets. We applied an algorithm based on SR -set to multi-attribute group decision making (MAGDM) to deal with uncertainty.

Rational approximation on $A^{\infty }(\varOmega )$

Let $\{U_i\}_{i\in I}$ be a family of bounded planar domains in $\mathbb C$. Here $A(\prod_{i\in I}U_i)$ stands for the set of continuous functions on $\prod_{i\in I}\overline{U_i}$ endowed with the product topology that are separately holomorphic on each

Control over the macrocyclisation pathway and product topology in a copper-templated catenane synthesis.

We report here that the product topology in a copper-templated catenane synthesis can be controlled by favouring a particular macrocyclisation pathway, offering an additional strategy for improving the efficiency of catenane formation. A linear [4]catenane was obtained by non-covalently modifying a flexible building block that favours the intra-ligand cyclisation.