Coarsest Topology Research Articles

Causal structure of spacetime and Scott topology

In this paper we establish the spacetime manifold as a partially ordered set via the casual structure. We show that these partially ordered sets are naturally continuous as a suitable way below relation can be established via the chronological order. We further consider those classes of spacetimes on which a lattice structure can be endowed by physically defining the joins and meets. By considering the physical properties of null geodesics on the spacetime manifold we show that these lattices are necessarily distributive. These lattices are then continuous as a result of the equivalence between the way below relation and chronology. This enables us to define the Scott topology on the spacetime manifold and describe it on an equal footing as any other continuous lattice. We further show that the Scott topology is a proper subset of Alexandroff topology, which must be the manifold topology for the strongly causal spacetimes, (and hence a coarser topology than Alexandroff). In the process we find some interesting results on the sobriety of these manifolds. We prove that they are necessarily not sober under the Scott topology but regain their sobriety under Alexandroff topology. We also define a dual Scott topology on these manifolds by endowing them with bicontinuous poset structure and show that the join of the Scott topology with the dual is the Alexandroff topology. We also discuss the previous works done in this topic and how the present work generalises those results to some extent.

Semi-Regularization Topological Spaces

If ( X , τ ) is a topological space, then the semi-regularization topology τ s on X of τ is the coarser topology on X generated by the family of all open domains of ( X , τ ) where a subset U is called an open domain if U = int(U). In this paper, we study the semi-regularity of some generated spaces and some properties of weaker version of normality of the semi-regularization space ( X , τ s ) of a space ( X , τ ).

ON d-I-proper function

In this paper,we introduce a new class of function called δ-I-proper function and explain some of propositions,theorems and some equivalent statements of this function . Key words and phrases: -I-cluster point,R-I-open set ,-I-open set , -I-continuous function,-I-closed function,-I-proper function . 1- Introduction: Let A be a subset of a topological space (X,T).The δ-interior of a subset A of X is the union of all regular open (R-open) sets of contained in A and is denoted by Int(A) [7] . The subset A is called δ-open if A=Int(A) (i.e. a set A is δ-open if it is the union of regular open sets )[7] .The complement of a δ-open set is called δ-closed . Alternatively , a set A(X,T) is called δ-closed if A=Cl(A), where Cl(A)={xX:Int(Cl(U))⋂A, UT and xU}[1].The family of all δ-open sets forms a topology on X and denoted by T .Since the intersection of two regular open sets is regular open , the collection of all regular open sets forms a base for a coarser topology TS than the original one T . A basic facts are that TS= T and every clopen set is regular open set [7] . In a topological space (X,T) , let I an ideal of subsets of X .An ideal is defined as a non-empty collection I of subsets of X satisfying two conditions : 1) If AI and BA , then BI ; 2) If AI and BI , then ABI .A topological (X,T) with an ideal I on X is called ideal topological space and denoted by (X,T,I) . For a subset AX , A={xX:U⋂AI , UT and xU} is called the local function of A with respect to I and T [3] .For each ideal topological space (X,T,I) , there exists a topology T finer than T , generated by (I,T)={U\I:UT and II}, but in general is not always a topology [2].Additionally , Cl(A)=AA defines a Kuratowiski closure for T and T T .

Tukey order and diversity of free Abelian topological groups

For a Tychonoff space X the free Abelian topological group over X , denoted A ( X ) , is the free Abelian group on the set X with the coarsest topology so that for any continuous map of X into an Abelian topological group its canonical extension to a homomorphism on A ( X ) is continuous. We show there is a family A of maximal size, 2 c , consisting of separable metrizable spaces , such that if M and N are distinct members of A then A ( M ) and A ( N ) are not topologically isomorphic (moreover, A ( M ) neither embeds topologically in A ( N ) nor is an open image of A ( N ) ). We show there is a chain C = { M α : α < c + } , of maximal size, of separable metrizable spaces such that if β < α then A ( M β ) embeds as a closed subgroup of A ( M α ) but no subspace of A ( M β ) is homeomorphic to A ( M α ) . We show that the character (minimal size of a local base at 0) of A ( M ) is d (minimal size of a cofinal set in N N ) for every non-discrete, analytic M , but consistently there is a co-analytic M such that the character of A ( M ) is strictly above d . The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an A ( X ) , and a connection with the family of compact subsets of an auxiliary space.

All adapted topologies are equal

A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

Taguchi design-assisted immobilization of Candida rugosa lipase onto a ternary alginate/nanocellulose/montmorillonite composite: Physicochemical characterization, thermal stability and reusability studies

Biomass from oil palm frond leaves (OPFL) is an excellent reservoir of lignocellulosic material which full potential remains untapped. This study aimed to statistically optimize the covalent immobilization of Candida rugosa lipase (CRL) onto a ternary support comprised of OPFL derived nanocellulose (NC) and montmorillonite (MMT) in alginate (ALG) (CRL-ALG/NC/MMT). The coarser topology and the presence of characteristic spherical globules in the field emission scanning electron micrographs and atomic force micrographs, respectively, supported the existence of CRL on ALG/NC/MMT. In addition, amide peaks at 3478 and 1640 cm−1 in the fourier transform infrared spectra affirmed that CRL was covalently bonded to ALG/NC/MMT. The optimized Taguchi Design-assisted immobilization of CRL onto ALG/NC/MMT (7 h of immobilization, 35℃, pH 5, 7 mg/mL protein loading) gave a production yield of 92.89 % of ethyl levulinate (EL), as proven by gas chromatography–mass spectrometric ([M] +m/z 144, C7H12O3), FTIR and nuclear magnetic resonance (CAS-539-88-8) data. A higher optimal reaction temperature (50℃) and the reusability of CRL-ALG/NC/MMT for up to 9 esterification cycles substantiated the appreciable structural rigidification of the biocatalyst by ALG/NC/MMT, which improved the catalytic activity and thermal stability of the lipase.

Epi-mild normality

AbstractA space (X, τ) is called epi-mildly normal if there exists a coarser topology τ′ on X such that (X, τ′) is Hausdorff (T2) mildly normal. We investigate this property and present some examples to illustrate the relationships between epi-mild normality and other weaker kinds of normality.

Epiregular topological spaces

A topological space (X, $$\tau $$ ) is called epiregular if there is a coarser topology $$\tau $$ $$^\prime $$ on X such that (X, $$\tau $$ $$^\prime )$$ is $$T_3$$ . We investigate this property and present some examples to illustrate the relationships between epiregular, epinormal, submetrizable, semiregular and almost regular.

A study of function space topologies for multifunctions

&lt;div class="page" title="Page 1"&gt;&lt;div class="layoutArea"&gt;&lt;div class="column"&gt;&lt;p&gt;&lt;span&gt;Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (&lt;/span&gt;&lt;span&gt;τ,μ&lt;/span&gt;&lt;span&gt;)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting. &lt;/span&gt;&lt;/p&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;

The Dirichlet problem at the Martin boundary of a fine domain

We adapt the Perron–Wiener–Brelot method of solving the Dirichlet problem at the Martin boundary of a Euclidean domain so as to cover also the Dirichlet problem at the Martin boundary of a fine domain U in Rn (n≥2) (i.e., a set U which is open and connected in the H. Cartan fine topology on Rn, the coarsest topology in which all superharmonic functions are continuous). It is a complication that there is no Harnack convergence theorem for so-called finely harmonic functions. We define resolutivity of a numerical function on the Martin boundary Δ(U) of U. Our main result Theorem 4.14 implies the corresponding known result for the classical case. We also obtain analogous results for the case where the upper and lower PWB-classes are defined in terms of the minimal-fine topology on the Riesz–Martin space U‾=U∪Δ(U) instead of the natural topology. The two corresponding concepts of resolutivity are compatible.

On weak completeness of products and direct sums in locally convex cones

We discuss the weak completeness of product and direct sum cones in the lower, upper and symmetric topologies. For the weak lower, upper and symmetric topology of a direct sum cone, there correspond bases of upper, respectively lower and symmetric, closed members in a coarser topology which leads us to investigate the weak completeness of direct sum cones.

Gap metrics for time-varying linear systems in a continuous-time setting

A recently proposed time-varying generalisation of Vinnicombe’s ν-gap is shown to have the properties of a metric on a class of causal linear continuous-time systems that admit normalised strong graph representations. An alternative time-varying gap metric is then defined in more general terms and a corresponding robust stability result is derived for linear feedback interconnections. Finally, it is shown that the time-varying gap and ν-gap metrics induce the same topology, this being the coarsest topology under which feedback stability and closed-loop sensitivity are both robust properties.

Between local connectedness and sum connectedness

A new generalization of local connectedness called Z-local connectedness is introduced. Basic properties of Z-locally connected spaces are studied and their place in the hierarchy of variants of local connectedness, which already exist in the literature, is elaborated. The class of Z-locally connected spaces lies strictly between the classes of pseudo locally connected spaces (Commentations Math. 50(2)(2010),183-199) and sum connected spaces ($\equiv$ weakly locally connected spaces) (Math. Nachrichten 82(1978), 121-129; Ann. Acad. Sci. Fenn. AI Math. 3(1977), 185-205) and so contains all quasi locally connected spaces which in their turn contain all almost locally connected spaces introduced by Mancuso (J. Austral. Math. Soc. 31(1981), 421-428). Formulations of product and subspace theorems for Z-locally connected spaces are suggested. Their preservation under mappings and their interplay with mappings are discussed. Change of topology of a Z-locally connected space is considered so that it is simply a locally connected space in the coarser topology. It turns out that the full subcategory of Z-locally connected spaces provides another example of a mono-coreflective subcategory of TOP which properly contains all almost locally connected spaces.

The topology of the class of functions representable by Carleman type formulae, duality and applications

We set D to be a simply connected domain and we consider exhaustion function spaces, X∞(D) with the projective topology (see §1). We show that the natural topology on the topological dual of X∞(D), (X∞(D))′, is the inductive topology. As a main application we assume that D has a Jordan rectifiable boundary ∂D, and M ⊂ ∂D to be an open analytic arc whose Lebesque measure satisfies 0 < m(M) < m(∂D). We prove a result for the dual of NHM (D), which is the class of holomorphic functions in D which are represented by Carleman formulae on M ⊂ ∂D. Furthermore we show that the Cauchy Integral associated to f ∈ NHM (D) is an element of NH 1 M (D). Lastly, we solve an extremal problem for the dual of NHM (D). 1. Projective and Inductive topologies We set D to be a bounded simply connected domain of the complex plane C and {Di}i=1, i = 1, 2 . . . to be an increasing sequence of bounded simply connected domains (i.e. Di ⊂ Di+1, i = 1, 2 . . .) such that D = ⋃ iDi with Di ⊂ D, i = 1, 2, . . ., ∂Di → ∂D in the sense that {∂Di}i eventually surrounds each compact subdomain of D. Such a sequence of domains {Di}i is called an exhaustion of D. Furthermore we set X(Di) to be a function space on Di with topology Ti. For simplicity in the symbolism write Xi for X(Di), and assume that each Xi carries the topology Ti for all i ∈ N. In the following we construct the projective limit associated with Xi and we provide it with the projective topology. For all i ≤ j, fi ∈ Xi, i, j = 1, 2, . . ., define the connecting maps (1.1) μij : Xj → Xi, i ≤ j, such that μij(fj) = fi is the restriction of fj on Di. In addition note that for all i, j, k = 1, 2 . . ., i ≤ j ≤ k, holds μik = μij ◦ μjk. We consider X∞ to be the subspace of ∏ iXi whose elements f = (f1, f2, . . .) satisfy the relation fi = μij(fj) for all i ≤ j. X∞ is called the projective limit of the family {Xi}i with respect to the mappings μij and is denoted by X∞ = lim ←− μij(Xi, Ti). We set μi to be the restriction to X∞ of the projection map pi of ∏ iXi onto Xi, i = 1, 2 . . ., and we give X∞ the projective topology T∞ with respect to the family {(Xi, Ti), μi}i. That is the coarsest topology on X∞ for which each of the mappings μi : X∞ → (Xi, Ti), i ∈ N, is continuous. An element fi ∈ Xi is called a representative of f ∈ X∞, if μi(f) = fi, i ∈ N. Note that each element of X∞ has a unique representative in each Xi, but that an element of Xi does not necessarily 1991 Mathematics Subject Classification. Primary:46A13, 30E20; Secondary:30D55, 30E25.

Linear feedback systems and the graph topology

It is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed-loop stability and closed-loop performance are robust properties. In earlier works, similar topological results were obtained by exploiting the existence of particular coprime-factor system representations, not known to exist in general. By contrast, the development here does not rely on any specific system representations. Systems are simply characterized as subspaces of norm bounded input-output pairs, and the analysis hinges on the underlying geometric structure of the feedback stabilization problem. Unlike other work developed within such a framework, fundamental issues concerning the causality of feedback interconnections are discussed explicitly. The key result of this paper concerns the difference between linear feedback interconnections, with identical controllers, in terms of a performance/robustness related closed-loop mapping. Upper and lower bounds on the induced norm of this difference are derived, allowing for possibly infinite-dimensional input-output spaces and time-varying behavior. The bounds are both proportional to the gap metric distance between the plants, which clearly demonstrates the gap to be an appropriate measure of the difference between open-loop systems from the perspective of closed-loop behavior. To conclude, an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting.

On the Topology Induced by Mixtures of Exponentials

It is shown that if X is a real vector space of uncountable dimension then the coarsest topology on X for which the function $$x \mapsto \smallint _{x*} e^{\xi (x)} d\mu (x)$$ is continuous whenever μ is a measure on the dual space X * integrating the integrands is strictly coarser than the finest locally convex topology. This is derived from an inequality relating the averages of such a 'mixture of exponentials' on the vertices and facet midpoints, respectively, of a 'generalized octahedron' in a finite-dimensional space (Lemma 1).

A Theorem on Fine Connectedness

We prove that Ω ∖ E is a p-fine domain whenever Ω ⊂ Rn is a p-fine domain, E ⊂ Rn is p-polar, and 1 < p ≤ n. By a p-fine domain we understand an open connected set in the p-fine topology, i.e. in the coarsest topology making all p-superharmonic functions continuous. As an application of our main result, we establish a general version of minimum principle.

Asymptotic values of fine continuous functions

The set of asymptotic values of a continuous function on the open unit disc in ℝ2 forms an analytic set, in the sense of being a continuous image of a Polish space (complete, separable metric space). This was proved in [9] by J. E. McMillan, who had earlier given versions of this result for holomorphic and meromorphic functions. We extend his method to the case of a function on the open unit ball of ℝn which is continuous merely in the fine topology, the coarsest topology making all subharmonic functions continuous. In particular, we use a version of McMillan's ingenious metric on a certain space of equivalence classes of asymptotic paths. McMillan also proved in [9] that the set of point asymptotic values of a continuous function in the unit disc forms an analytic set. We use a modification of the McMillan metric to extend this result to fine continuous functions in the unit ball and deduce that the set of boundary points of the unit ball at which the function has an asymptotic value forms an analytic set.

Convergence structures induced by scales

By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τ μ . The following convergence structures on L are compared with each other: 1. (i) order convergence, 2. (ii) convergence in the order topology, 3. (iii) τ μ -convergence, 4. (iv) ρ μ -convergence, where ρ μ ( x, y) = μ( x ∨ y) − μ( x ∧ y). We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρ μ -convergence and with τ μ -convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρ μ -convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τ μ -convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τ μ -convergence iff μ is continuous and L is algebraic (or atomic and meet-continuous).

A Conjecture on the Hall Topology for the Free Group

The Hall topology for the free group is the coarsest topology such that every group morphism from the free group onto a finite discrete group is continuous. It was shown by M. Hall Jr. that every finitely generated subgroup of the free group is closed for this topology. We conjecture that if H1, H2, ..., Hn are finitely generated subgroups of the free group, then the product H1H2... Hn is closed. We discuss some consequences of this conjecture. First, it would give a nice and simple algorithm to compute the closure of a given rational subset of the free group. Next, it implies a similar conjecture for the free monoid, which, in turn, is equivalent to a deep conjecture on finite semigroup, for the solution of which J. Rhodes has offered $100. We hope that our new conjecture will shed some light on the Rhodes's conjecture.