Arbitrary Topological Space Research Articles

Quantum Fibrations: Quantum Computation on an Arbitrary Topological Space

Quantum Fibrations: Quantum Computation on an Arbitrary Topological Space

Domination of nonlinear semigroups generated by regular, local Dirichlet forms

AbstractIn this article we study perturbations of local, nonlinear Dirichlet forms on arbitrary topological measure spaces. As a main result, we show that the semigroup generated by a local, regular, nonlinear Dirichlet form $${\mathcal {E}}$$ E dominates the semigroup generated by another local functional $${\mathcal {F}}$$ F if, and only if, $${\mathcal {F}}$$ F is a specific zero order perturbation of $${\mathcal {E}}$$ E . On the way, we prove a nonlinear version of the Riesz–Markov representation theorem, we define an abstract boundary of a topological measure space, and apply the notion of nonlinear capacity. The main result helps to classify the perturbations that lie between Neumann and Dirichlet boundary conditions.

Best proximity results for p-proximal contractions on topological spaces

In this article, we investigate some sufficient conditions for the existence and uniqueness of best proximity points for the topological p-proximal contractions and p-proximal contractive mappings on arbitrary topological spaces. Moreover, our results are illustrated by a few numerical examples and they generalize some known results in the literature.

A characterization of productive cellularity

We investigate the notion of productive cellularity of arbitrary posets and topological spaces. Particularly, by working with families of antichains ordered with reverse inclusion, we give necessary and sufficient conditions to determine whether a poset or a topological space is productively ccc.

Cohomology of generalized Dold spaces

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e., the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where X,S are arbitrary topological spaces. The resulting space P(S,X) will be called a generalized Dold space. When S and X are CW complexes satisfying certain natural requirements, we obtain a CW-structure on P(S,X). Under certain further hypotheses, we determine the mod 2 cohomology groups of P(S,X). We determine the Z2-cohomology algebra when X is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over P(S,X) associated to σ-conjugate complex bundles over X when the S is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds.

Associative algebra twisted bundles over compact topological spaces

For the associative algebra A(g) of an infinite-dimensional Lie algebra g, we introduce twisted fiber bundles over arbitrary compact topological spaces. Fibers of such bundles are given by elements of algebraic completion of the space of formal series in complex parameters, sections are provided by rational functions with prescribed analytic properties. Homotopical invariance as well as covariance in terms of trivial bundles of twisted A(g)-bundles is proven. Further applications of the paper's results useful for studies of the cohomology of infinite-dimensional Lie algebras on smooth manifolds, K-theory, as well as for purposes of conformal field theory, deformation theory [10,14,15], and the theory of foliations are mentioned.

Intrinsic characterizations of C-realcompact spaces

<pre>c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.</pre>

Absolutes and n -H-closed spaces

In this paper the investigation of n -H-closed spaces that was started by Basile et al . (2019) is continued for every n ∈ ω, n ≥ 2. In particular, starting with the relationship between the absolute space PX of an arbitrary topological space X , reported by Ponomarev and Shapiro (1976) and introduced by Blaszczyk (1975, 1977), Ul'yanov (1975a,b) and Shapiro (1976), it is shown that the absolute PX is n -H-closed if and only if X is n -H-closed. For an arbitrary space X , a β-like extension (β for the Stone-Cech compactification) Y is constructed for the semiregularization PX(s) of the absolute PX such that Y is a compact, extremally disconnected, completely regular (but not necessarily Hausdorff) extension of PX(s) , and PX(s) is C* -embedded in Y . The definition of the Fomin extension σ X for a Hausdorff space X (Porter and Woods 1988) is extended to an arbitrary space X and σ X X is shown to be homeomorphic to the remainder Y PX(s) . A similar result is established when X is an n -Hausdorff space defined by Basile et al. (2019). Further, we give a cardinality bound for any n -Hausdorff space X and show that the inequality |X| ≤ 2^χ( X ) for an H-closed space X proved by Dow and Porter (1982) can be extended to n -H-closed spaces.

θ s -open sets and θ s -continuity of maps in the product space

In this study, the concept of \(\theta_s\)-open set is introduced. The topology formed by \(\theta_s\)-open sets is strictly finer than the topology formed by \(\theta\)-open sets but is not comparable with the topology formed by \(\omega_\theta\)-open sets. Related concepts such as \(\theta_s\)-open and \(\theta_s\)-closed functions, \(\theta_s\)-continuous function, \(\theta_s\)-connected space, and some versions of separation axioms are defined and characterized. Finally, the concept of \(\theta_s\)-continuous function from an arbitrary topological space into the product space is investigated further.

Generalization and new proof for almost everywhere convergence to imply local convergence in measure

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize the theorem for the case where the codomain is a separable metric space and for the case where the limiting map is constant and the codomain is an arbitrary topological space.

Gateaux Differentiability Revisited

We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided.

On topological soft sets

Abstract In this paper, we have established topological soft sets over generalized topological spaces and topological spaces, and studied its structural properties. We have derived a topological soft set in any given topological space, and from this point of view, we have given necessary and sufficient condition for homeomorphic Alexandroff spaces using topological soft set technique. At last, we have derived a topological soft set using closed sets in any topological space and we have given necessary and sufficient condition for arbitrary homeomorphic topological spaces using them.

A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

AbstractThe Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

Isomorphisms of Semirings of Continuous Nonnegative Functions and the Lattices of Their Subalgebras

Let $$S$$ be the semifield $$\mathbb{R}_{+}$$ with zero of nonnegative real numbers or the semifield $$\mathbb{P}$$ of positive real numbers. The set of all continuous functions $$f\colon X\to S$$ with pointwise operations of addition and multiplication of functions defined on an arbitrary topological space $$X$$ forms the semiring $$C(X,S).$$ By a subalgebra we mean a nonempty subset $$A$$ of $$C(X,S)$$ such that $$f+g,fg,rf\in A$$ for any $$f,g\in A$$ and any $$r\in S.$$ For arbitrary topological spaces $$X$$ and $$Y,$$ we describe isomorphisms of the semirings $$C(X,S)$$ and $$C(Y,S)$$ and isomorphisms of the lattices of their subalgebras (subalgebras with unity).

Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension

Taking the covering dimension dim as notion for the dimension of a topological space, we first specify the number zdimT0(n) of zero-dimensional T0-spaces on {1,…,n} and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,…,n} by means of two mappings po and P that yield the number po(n) of partial orders on {1,…,n} and the set P(n) of partitions of {1,…,n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdimT0(n) and modify one for P in such a way that it computes zdimT0(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdimT0(1) to zdimT0(n) and the Stirling numbers of the second kind S(n, 1) to S(n, n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdimT0(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.

Ordinal compactness

We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities. The most general form depends on two ordinal parameters. Ordinal compactness turns out to be a much more varied notion than cardinal compactness. We prove many nontrivial results of the form ?every [?,?]-compact topological space is [?',?']-compact?, for ordinals ?,?, ?'and ?' while only trivial results of the above form hold, if we restrict to regular cardinals. Counterexamples are provided showing that many results are optimal. Many spaces satisfy the very same cardinal compactness properties, but have a broad range of distinct behaviors, as far as ordinal compactness is concerned. A much more refined theory is obtained for T1 spaces, in comparison with arbitrary topological spaces. The notion of ordinal compactness becomes partly trivial for spaces of small cardinality.

Best proximity point theorems in topological spaces

Let A, B be nonempty subsets of a metric space X and $$f:A\rightarrow B$$ be a mapping. A point $$x_0\in A$$ is a best proximity point of f if $$d(x,f(x))=\mathrm{dist}(A,B):=\{d(a,b):a\in A,b\in B\}$$. It is worth mentioning that the metric function d plays a vital role in defining the notion of best proximity points. In this manuscript, we introduced a notion of best proximity points in arbitrary topological spaces and established few best proximity point theorems. Our main result generalizes the well-known Edelstein’s fixed point theorem for contractive mappings.

On the class of weakly almost contra-$T^*$-continuous functions

The aim of this paper is to introduce and investigate a new class of functions called weakly almost contra-$T^*$-continuity which is defined as a function from an operator topological space $(X, \tau, T)$ into an arbitrary topological space $(Y, \delta)$. Furthermore, some new characterizations, several basic propositions are proved and some relevant counterexamples are provided.

Isomorphisms of Lattices of Subalgebras of Semifields of Positive Continuous Functions

We consider the lattice of subalgebras of a semifield U(X) of positive continuous functions on an arbitrary topological space X and its sublattice of subalgebras with unity. We prove that each isomorphism of the lattices of subalgebras with unity of semifields U(X) and U(Y) is induced by a unique isomorphism of the semifields. The same result holds for lattices of all subalgebras excluding the case of the double-point Tychonoff extension of spaces.

Formalization of the Fundamental Group in Untyped Set Theory Using Auto2

We present a new framework for formalizing mathematics in untyped set theory using auto2. Using this framework, we formalize in Isabelle/FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.