Showing papers on "Topological space published in 1986"
TL;DR: In this paper, the authors prove existence theorems for the solutions of functional equations that arise in dynamic programming, based on fixed point theoremms for linear topological spaces.
Abstract: On the basis of fixed point theorems for linear topological spaces the authors prove existence theorems for the solutions of functional equations that arise in Dynamic Programming.
62 citations
TL;DR: In this paper, a sufficient condition for a convex coneC in a Hausdorff topological linear space is given in order to ensure the existence of cone-maximal points.
Abstract: A sufficient condition for a convex coneC in a Hausdorff topological linear space is given in order to ensure the existence of cone-maximal points. The condition becomes a necessary one in a topological linear space with a countable local base, that is, if the space is pseudometrizable. The paper extends known results to infinite dimensions and we answer Corley’s question in the affirmative with the exception of a pathological case.
45 citations
TL;DR: In this article, the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies is discussed, and necessary and sufficient conditions for the existence of a solution are given.
43 citations
TL;DR: In this paper, the authors point out the link between Lowen's fuzzy topologies and topological space objects in the topos L-SET and obtain the following theorem: Lowening's fuzzy structures are the external version of the internal topologies in L-set.
43 citations
31 Aug 1986
TL;DR: An algorithm is presented for automatically converting data representing unambiguous, three-dimensional objects in wire-frame form with curvilinear edges into a boundary representation, an important extension to a previously published algorithm based on graph theory and topology.
Abstract: An algorithm is presented for automatically converting data representing unambiguous, three-dimensional objects in wire-frame form with curvilinear edges into a boundary representation. The method is an important extension to a previously published algorithm based on graph theory and topology. The new method automatically detects and resolves anomalies, such as necks which may appear to be faces, that formerly required human intervention. The topological basis for the solution to this problem is given along with a description of what topological properties a well defined three-dimensional object should have. An implementation has been coded and examples of results are included.
42 citations
TL;DR: The concept of almost fuzzy continuous mapping from a fuzzy topological space into another is introduced and discussed and the benefits of this approach are discussed.
37 citations
TL;DR: In this article, a generalization of the compact-open topology is defined for a space of utility functions with different choice sets, and a homeomorphism between this space and the space of preference relations, with the latter given a certain topology coarser than closed convergence.
35 citations
TL;DR: In this paper, the authors generalize the classic theorems of Eilenberg and Debreu on the existence of continuous order-preserving transformations on ordered topological spaces and prove them in a different way.
Abstract: The object of this paper is to generalize the classic theorems of Eilenberg and Debreu on the existence of continuous order-preserving transformations on ordered topological spaces and to prove them in a different way. The proof of the theorems is based on Nachbin's generalization to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.
35 citations
TL;DR: For spaces with infinitely many nonempty derivatives a strong negative theorem is obtained and it is possible to partition the pairs of rationals into countably many pieces so that every homeomorph of the rationals contains a pair from every piece.
32 citations
TL;DR: In this paper, the authors studied the completeness properties of the space C k (X ) of continuous real-valued functions on a topological space X, where the function space has the compact-open topology.
32 citations
TL;DR: In this paper, the authors established conditions under which an interval order > on a topological space X can be represented by a pair of real-valued functions u, v on X, in the sense that x > y if and only if u(x)>v(y).
TL;DR: Topological invariants such as the singular homology groups of Σn,m are explicitly computed and they are shown to coincide with those of a certain Grassmann manifold.
Abstract: This paper deals with the algebraic topology of the space Σ
n,m
of complex reachable linear dynamical systems. Topological invariants such as the singular homology groups of Σ
n,m
are explicitly computed and they are shown to coincide with those of a certain Grassmann manifold. From this some new results on the topology of rational transfer matrices with fixed McMillan degree are obtained.
TL;DR: This paper defines separation functions which 'measure' the degree to which points in a given space can be separated in a T"0-, T"1- or T"2-like fashion.
TL;DR: In this article, sufficient conditions are given in order to ensure the existence of a sequence of strongly consistent estimators of unknown parameters in a nonlinear regression model, where the parameter space is assumed to be any separable, completely regular topological space.
Abstract: Sufficient conditions are given in order to ensure the existence of a sequence of strongly consistent estimators of unknown parameters in a nonlinear regression model. The primary difference between this and earlier work is in the generality of the parameter space. Indeed, the parameter space is assumed to be any separable, completely regular topological space; in particular, this includes all separable metric spaces.
TL;DR: In this article, the authors studied the topological properties of two kinds of fine topologies on the space C ( X, Y ) of all continuous functions from X into Y.
Abstract: This paper studies the topological properties of two kinds of fine topologies on the space C ( X , Y ) of all continuous functions from X into Y .
TL;DR: For a quotient-reflective subcategory as mentioned in this paper of the category Topof topological spaces, the following "diagonal theorem" is proved: a topological space (X,τ) belongs to the diagonal Δxis (τ×τ)A-closed, where, for X,τ, the coarsest topology on X has as closed subsets all the equalizers of pairs of continuous maps with codomain in A.
Abstract: For a quotient-reflective subcategoryAof the category Topof topological spaces the following «diagonal theorem» is proved: a topological space (X,τ)belongs toAiff the diagonal Δxis (τ×τ)A-closed, where, for (X, ρ) e Top, σAdenotes the coarsest topology on X which has as closed subsets all the equalizers of pairs of continuous maps with codomain inA.Furthermore an explicit description of τAfor several quotient reflective subcategories defined by means of properties of subspaces is given. It is shown that one of them is not co-(well-powered).
TL;DR: Bell and Ginsburg as discussed by the authors showed that the topological space M(P) is compact if and only if, for each x ∊ P, there is a finite subset C(x) of P all of whose elements are non-comparable to x and such that {x} ∪ C (x) meets each maximal chain.
Abstract: This paper is inspired by two apparently different ideas. Let P be an ordered set and let M(P) stand for the set of all of its maximal chains. The collection of all sets of the form and where x ∊ P, is a subbase for the open sets of a topology on M(P). (Actually, it is easy to check that the B(x) sets themselves form a subbase.) In other words, as M(P) is a subset of the power set 2 |p| of P, we can regard M(P) as a subspace of 2 |p| with the usual product topology. M. Bell and J. Ginsburg [1] have shown that the topological space M(P) is compact if and only if, for each x ∊ P, there is a finite subset C(x) of P all of whose elements are noncomparable to x and such that {x} ∪ C(x) meets each maximal chain.
01 Feb 1986
TL;DR: In this article, the authors give an example of a weakly infinite-dimensional space X such that the product X X B of X with a certain subspace B of the space of irrationals is strongly infinite.
Abstract: We give an example of a weakly infinite-dimensional space X such that the product X x B of X and a subspace B of the irrationals is strongly infinite-dimensional; under the assumption of the Continuum Hypothesis, B can be the irrationals. This example answers a question of Addis and Gresham [AG]. 1. Terminology and notation. All spaces under discussion are metrizable and separable. Our terminology follows [AP and E]. We denote by I the real interval [0, 1], by C the usual Cantor set in I and by I' the Hilbert cube. We denote the space of the irrational numbers from I by P and the rational numbers from I by Q. A space X is weakly infinite-dimensional [AP, Chapter 10, ??4-7] if for every sequence {(A1, B1), (A2, B2), ... } of pairs of closed disjoint subsets of X there are partitions L, in X between Ai and Bi such that nfo 1L, = 0. Otherwise, X is strongly infinite-dimensional. A space X is called a C-space if for every sequence 1, .. . of open covers of X there exists a sequence q1, q&2,*.. of families of open subsets of X such that, for (i) the members of q&i are pairwise disjoint, (ii) each member of q&, is contained in a member of 9i, (ii) the union U7 10&i covers X. The notion of C-space was introduced by W. Haver in [H] for metric space and by D. Addis and J. Gresham in [AG] for general topological spaces. LEMMA 1 [AG]. Every C-space is weakly infinite-dimensional. 2. Results. The aim of this note is to construct the following examples. EXAMPLE 1. There exists a weakly infinite-dimensional space X such that the product X X B of X with a certain subspace B of the space of irrationals is strongly infinite-dimensional. Moreover, X is a C-space while X X B is not a C-space. EXAMPLE 2. Under the assumption of the Continuum Hypothesis there exists a weakly infinite-dimensional space X such that the product X X P of X with the space of irrationals P is strongly infinite-dimensional. Moreover, X is a C-space while X X P is
TL;DR: Concepts of semi-Ti (i = 0, 1, 2) spaces and semi-Ri spaces and some fuzzy topological properties are investigated under the above mentioned axioms.
31 Aug 1986
TL;DR: The Major-minor color space is developed, which has a topology and representation that lends itself to simple anti-aliasing computations between elements of an arbitrary set of colors in an inexpensive frame store.
Abstract: The power of a color space to perform well in interpolation problems such as anti-aliasing and smooth-shading is dependent on the topology of the color space as well as the number of elements it contains.We develop the Major-minor color space, which has a topology and representation that lends itself to simple anti-aliasing computations between elements of an arbitrary set of colors in an inexpensive frame store.
IBM1
TL;DR: The languages appearing in the theory of ω-automata are shown to be closely related to functionally closed sets (continuous inverse images of the set {0}).
TL;DR: In this article, the Borel-inseparable pairs of coanalytic sets in Polish spaces are studied, where the class of analytic sets is closed under countable unions and intersections, images and preimages by Borel measurable functions, and projections.
Abstract: The purpose of this paper is to give some natural examples of Borel-inseparable pairs of coanalytic sets in Polish spaces. A Polish space is a topological space homeomorphic to a separable complete metric space. In this paper, all spaces are uncountable Polish spaces. A pointset is analytic (or ) if it is the continuous image of a Borel set (in any space), or equivalently, the projection of a Borel set, and is coanalytic (or ) if it is the complement of an analytic set. The class of analytic sets is closed under countable unions and intersections, images and preimages by Borel measurable functions, and projections; it is not closed under complements, hence there is an analytic set which is not Borel.
TL;DR: In this paper, it was shown that a topological space admits a unique quasi-uniformity if and only if every interior-preserving open collection of the space is finite.
Abstract: Abstract We show that a topological space X admits a unique quasiuniformity if and only if every interior-preserving open collection of X is finite.
TL;DR: In this article, the authors characterize and study the class of topological spaces that admit the coarsest quasi-proximity and show that the two classes coincide in the super-sober spaces.
Abstract: We characterize and study the class of topological spaces that admit a coarsest quasi-proximity. This class of spaces generalizes the class of locally compact spaces in a natural way. It follows from our results that the two classes coincide in the class of super-sober spaces.
TL;DR: In this article, Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological space is shown to be order-isomorphic to the real numbers.
Abstract: Fleischer proved that a linearly ordered set that is separable in its order topology and has countably many jumps is order-isomorphic to a subset of the real numbers. The object of this paper is to extend Fleischer's result and to prove it in a different way. The proof of the theorem is based on Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.
TL;DR: In this paper, the basic difficulties in the construction of quantum topological transition theory are connected with a necessity to introduce a new non-local interaction defined on a space of topological states.
Abstract: Results of a realisation of the topological transitions hypothesis are presented. The basic difficulties in the construction of quantum topological transition theory are connected with a necessity to introduce a new non-local interaction defined on a space of topological states. So the general method of construction and study of topological transitions classical models is formulated as a necessary step towards a corresponding quantum description. Their local properties, including an asymptotic behaviour in the neighbourhood of the transition, are studied and applications to problems of gravitation and cosmology are given. The method used is shown to lead to a scalar-tensor theory of topological transitions. Different variants of this theory and its main features are discussed.
TL;DR: The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly 2λ, and every completely regular space of weight λ is embeddable in a space of this type as discussed by the authors.
TL;DR: In this article, it was shown that the complete varieties, the compact (analytic) spaces, and the compact manifolds characterize the gross topoi of algebraic, analytic, and differential geometry.
Abstract: It is well known that compact topological spaces are those space K for which given any point x 0 in any topological space X , and a neighborhood H of the fibre -1 { x 0 } K X X , then there exists a neighborhood U of x 0 such that -1 U H . If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K , we have ( -1 A B )= A B . We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.