Showing papers on "Topological space published in 1991"

Point-set topological spatial relations

Abstract: Practical needs in geographic information systems (GIS) have led to the investigation of formal and sound methods of describing spatial relations. After an introduction to the basic ideas and notions of topology, a novel theory of topological spatial relations between sets is developed in which the relations are defined in terms of the intersections of the boundaries and interiors of two sets. By considering empty and non-empty as the values of the intersections, a total of sixteen topological spatial relations is described, each of which can be realized in R 2. This set is reduced to nine relations if the sets are restricted to spatial regions, a fairly broad class of subsets of a connected topological space with an application to GIS. It is shown that these relations correspond to some of the standard set theoretical and topological spatial relations between sets such as equality, disjointness and containment in the interior.

Reasoning about Binary Topological Relations

Abstract: A new formalism is presented to reason about topological relations. It is applicable as a foundation for an algebra over topological relations. The formalism is based upon the nine intersections of boundaries, interiors, and complements between two objects. Properties of topological relations are determined by analyzing the nine intersections to detect, for instance, symmetric topological relations and pairs of converse topological relations. Based upon the standard rules for the transitivity of set inclusion, the intersections of the composition of two binary topological relations are determined. These intersections are then matched with the intersections of the eight fundamental topological relations, giving an interpretation to the composition of topological relations.

A topological approach to digital topology

Abstract: (1991). A Topological Approach to Digital Topology. The American Mathematical Monthly: Vol. 98, No. 10, pp. 901-917.

Finitary substitute for continuous topology

Abstract: Finite topological spaces are combinatorial structures that can serve as replacements for, or approximations to, bounded regions within continuous spaces such as manifolds. In this spirit, the present paper studies the approximation of general topological spaces by finite ones, or really by “finitary” ones in case the original space is unbounded. It describes how to associate a finitary spaceF with any locally finite covering of aT 1-spaceS; and it shows howF converges toS as the sets of the covering become finer and more numerous. It also explains the equivalent description of finite topological spaces in order-theoretic language, and presents in this connection some examples of posetsF derived from simple spacesS. The finitary spaces considered here should not be confused with the so-called causal sets, but there may be a relation between the two notions in certain situations.

Topology and category theory in computer science

Abstract: A.W. Roscoe: Topology, computer science and the mathematics of convergence Stepen Blamey: The soundness and completeness of axioms for CSP processes Geoff Barrett & Michael Goldsmith: Classifying unbounded nondeterminism in CSP Michael W. Mislove: Algebraic posets, algebraic cpo's and models of concurrency J.W. de Bakker & J.J.M.M. Rutten: Concurrency semantics based on metric domain equations Marta Z. Kwiatkowska: On topological characterization of behavioral properties J.D. Lawson: Order and strongly sober compactifications Michael B. Smyth: Totally bounded spaces and compact ordered spaces as domains of computation Dieter Spreen: A characterization of effective topological spaces II Klaus E. Grue: The importance of cardinality, separability, and compactness in computer science with an example from numerical signal analysis T.Y. Kong: Digital topology: a comparison of the graph-based and topological approaches D. Girault-Beauquier & M. Nivat: Tiling the plane with one tile Narcisco Marti-Oliet & Jose Meseguer: An algebraic axiomatization of linear logic models Joseph A. Goguen: Types as theories.

Counting finite posets and topologies

Abstract: A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n≤11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n≤14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n≤11 and all k, and then the numbers of all topologies on n≤14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 1023.

On Rings of Continuous Functions on Topological Spaces

Abstract: This paper is related to studies byM.H.Stone [2] and to the above paper by G.E. Shilov. In contrast to the latter, we consider the ring of continuous functions on a topological space as a purely algebraic object without defining any topological relations in it. It turns out that in the case of bicompact spaces, considered by M.H. Stone, and also in some much more general cases, even the purely algebraic structure of the ring of continuous functions determines the topological space to within a homeomorphism.

An Introduction to Topology and Homotopy

Abstract: Part I: Topology. Preliminaries. Metric topology. Topological equivalence. Topological spaces. Construction techniques. Connectedness. Compactness. Part II: Homotopy. Fundamental group. Homotopy. Group theory. Calculation of Pi1 surfaces. Covering spaces. CW complexes.

A general theory of structure spaces with applications to spaces of prime ideals

Abstract: A structure space is a quadrupleX=(X, d, A, P), where for some setR, X ⊂A=2 R ,d:X×X →A is defined byd(I, J)=J−I, andP is the family of cofinite subsets ofR Forr e P, I e X, N r (I)={J e X: d(I, J) ⊂r},To(X)={Q ⊂X: if x e Q there is anr e P such thatN r (x) $$ \subseteq$$ Q} ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology Replacing d byd *, whered * (I, J)=d(J, I), or byd s, whered s (I, J)=d(I, J) ∪d * (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X *) andsymmetric topology To(X s ) The latter is always a zero-dimensional Hausdorff space When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X s ) is usually called thepatch topology Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring

Bicompleteness of the fine quasi-uniformity

Abstract: A characterization of the topological spaces that possess a bicomplete fine quasi-uniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable T1-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T1-spaces that do not admit a bicomplete quasi-uniformity.We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober.

Separated and Weakly Separated Subspaces of Topological Spaces

Abstract: Summary. A new concept of weakly separated subsets and subspaces of topological spaces is described in Mizar formalizm. Based on [1], in comparison with the notion of separated subsets (subspaces), some properties of such subsets (subspaces) are discussed. Some necessary facts concerning closed subspaces, open subspaces and the union and the meet of two subspaces are also introduced. To present the main theorems we first formulate basic definitions. Let X be a topological space. Two subsets A1 and A2 of X are called weakly separated if A1 \A2 and A2 \A1 are separated. Two subspaces X1 and X2 of X are called weakly separated if their carriers are weakly separated. The following theorem contains a useful characterization of weakly separated subsets in the special case when A1 ∪ A2 is equal to the carrier of X. A1 and A2 are weakly separated iff there are such subsets of X, C1 and C2 closed (open) and C open (closed, respectively), that A1 ∪ A2 = C1 ∪ C2 ∪ C, C1 ⊂ A1, C2 ⊂ A2 and C ⊂ A1 ∩ A2. Next theorem divided into two parts contains similar characterization of weakly separated subspaces in the special case when the union of X1 and X2 is equal to X. If X1 meets X2, then X1 and X2 are weakly separated iff either X1 is a subspace of X2 or X2 is a subspace of X1 or there are such open (closed) subspaces Y1 and Y2 of X that Y1 is a subspace of X1 and Y2 is a subspace of X2 and either X is equal to the union of Y1 and Y2 or there is a(n) closed (open, respectively) subspace Y of X being a subspace of the meet of X1 and X2 and with the property that X is the union of all Y1, Y2 and Y . If X1 misses X2, then X1 and X2 are weakly separated iff X1 and X2 are open (closed) subspaces of X. Moreover, the following simple characterization of separated subspaces by means of weakly separated ones is obtained. X1 and X2 are separated iff there are weakly separated subspaces Y1 and Y2 of X such that X1 is a subspace of Y1, X2 is a subspace of Y2 and either Y1 misses Y2 or the meet of Y1 and Y2 misses the union of X1 and X2.

Strong shape for topological spaces

Abstract: Strong shape equivalences for topological spaces are introduced in a way which generalizes easily to inverse systems of topological spaces. Each space is then mapped via a strong shape equivalence into a fibrant inverse system of ANRs. This leads naturally to defining the strong shape category SSh for topological spaces. Other descriptions of SSh are also provided.

To-OBJECTS AND SEPARATED OBJECTS IN TOPOLOGICAL CATEGORIES

Abstract: Several well-known generalizations of the usual topological To-axiom to topological categories are compared. It is shown that they lead to two different concepts: To and separatedness. The resulting sub-categories are characterized for some important categories of convergence spaces. They provide natural examples disproving the equivalence of the two concepts. Necessary and sufficient conditions are given for the To-objects and separated objects to coincide.

On the Linear Dimension of Topological Vector Spaces

Abstract: Two topological vector spaces E and E’ are said to be isomorphic if it is possible to establish a one-to-one linear bicontinuous correspondence between them.