Showing papers on "Topological space published in 1979"

Fuzzy digital topology

Abstract: Topological relationships among parts of a digital picture, such as connectedness and surroundedness, play an important role in picture analysis and description. This paper generalizes these concepts to fuzzy subsets, and develops some of their basic properties.

The topology of spaces of rational functions

Abstract: where a, and b t are complex numbers, defines a continuous map of degree n from the Riemann sphere S2=CtJ co to itself. I f the coefficients (a 1 .. . . . an; bl . . . . . bn) va ry cont inuously in C 2\" the m a p ] varies continuously providing the polynomials p and q have no root in common; but the topological degree of the map / jumps when a root of p moves into coincidence with a root of q. Let F* denote the open set of (~\" consisting of pairs of monic polynomials (p, q) of degree n with no common root. F* is the complement of an algebraic hypersurface, the \"resul tant locus\", in (~2~. On the other hand it can be identified with a subspace of the space M~* of maps $ 2 ~ S 2 which take c~ to 1 and have degree n. I n this paper I shall prove tha t when n is large the 2n-dimensional complex var ie ty F* is a good approximat ion to the homotopy type of the space M*, or, more precisely

Convergence in fuzzy topological spaces

Abstract: In the first paragraph we study filters in the lattice IX, where I is the unitinterval and X an arbitrary set. The main result of this section is a characterization of minimal prime filters in IX containing a given filter in IX by means of ultrafilters on X. In the second paragraph we apply the results of the previous section to define convergence in a fuzzy topological space which enables us to characterize fuzzy compactness and fuzzy continuity.

On epireflective subcategories of topological categories

Abstract: In this paper the lattice of all epireflective subcategories of a topological category is studied by defining the T 0 -objects of a topological category. A topological category is called universal iff it is the bireflective hull of its T 0 -objects. Topological spaces, uniform spaces, and nearness spaces form universal categories. The lattice of all epireflective subcategories of a universal topological category splits into two isomorphic sublattices. Some relations and consequences of this fact with respect to cartesian closedness and simplicity of epireflective subcategories are obtained.

The free topological group over the rationals

Abstract: In this paper we investigate the topological structure of the Graev free topological group over the rationals. We show that this free group fails to be a k -space and fails to carry the weak topology generated by its subspaces of words of length less than or equal to n . As tools in this investigation we establish some properties of net convergence in free groups and also some properties of certain canonical maps which are closely related to the topological structure of free groups.

On “Ultraproducts in topology”

Abstract: We correct an error in the above-mentioned paper and provide a solution to an open problem contained therein.

The total negation of a topological property

Abstract: The central theme in this paper is the uniform generation of new topological properties from old. Two of the best known properties obtained in this way are total disconnectedness (deriving from connectedness) and scatteredness (deriving from perfectness, i.e. having no isolated points). A third property, lesser known but interesting in its own right, is pseudofiniteness (the cf-spaces studied in [8], [9], [10], [12]) or the class of spaces whose compact subsets are finite. This last-mentioned property derives from compactness in the manner we will explore here. In general, given a class K of topological spaces (K is closed under homeomorphism) we define the class Anti (K) in such a way that "totally disconnected" is co-extensive with "Anti (connected)" and so on. The "anti-property" of most interest to us here is pseudofiniteness which we henceforth relabel "anticompactness". We will also be interested in related anti-properties (Anti (sequentially compact), Anti (LindelSf), etc.) but they will recieve secondary emphasis. The general behavior of the operation Anti (.) itself will occupy some of our attention. However at this stage there are many more questions than answers, so our general treatment will be sketchy, serving mainly to tie together ideas which otherwise may appear to be unrelated. Our set-theoretic conventions are as follows: (i) o denotes the ath infinite initial ordinal, where a is any ordinal. Since we assume the Axiom of Choice throughout, we identify % with the cardinal I. o o0. (ii) An ordinal a is the set of its predecessors. Greek letters near the beginning of the alphabet will usually denote ordinals, while the letters K, A, t will be reserved for cardinals. (iii) The ordinal successor of a is a + 1 a tO {a}, the cardinal successor of K is /. (iv) IfA is any set P(A) denotes the power set of A. (v) B A is the set of all maps f: A --* B. The cardinality of A is written IAI. (vi) If is a cardinal then exp()=12Kl=lP()l, exp(o)is usually denoted by c. (vii) The cartesian product of a family (Ai: i I) of sets is denoted 1-I Ai. If Ai A for all i I then the set A will also at times be denoted 1-I (A). Further notations will be introduced as they arise in the discussion. The referee’s kind suggestions regarding exposition are gratefully acknowledged.

Connections between convergence and nearness

Abstract: Generalizing usual filter convergence, we introduce stack convergence. This convergence contains nearly each essential concept of generalizing topological and uniform spaces. In particular, the belonging category SCO includes as well the topological spaces as the prenearness spaces. Thus we have solved a problem of Herrlich (1974) and have proved: Convergence contains nearness.

A Note on the Normal Moore Space Conjecture

Abstract: F. B. Jones (1937) conjectured that every normal Moore space is metrizable. He also denned a particular kind of topological space (now known as Jones' spaces), proved that they were all non-metrizable Moore spaces, but was unable to decide whether or not Jones’ spaces are normal. J. H. Silver (1967) proved that a positive solution to Jones’ conjecture was not possible, and W. Fleissner (1973) obtained an alternative proof by showing that it is not possible to prove the non-normality of Jones’ spaces. These results left open the possibility of resolving the questions from the GCH. In this paper we show that if CH be assumed, then Jones’ spaces are not normal (Devlin, Shelah, independently) and that the GCH does not lead to a positive solution to the Jones conjecture (Shelah). A brief survey of the progress on the problem to date is also included.

Recursive constructions in topological spaces

Abstract: We study topological constructions in the recursion theoretic framework of the lattice of recursively enumerable open subsets of a topological space X. Various constructions produce complemented recursively enumerable open sets with additional recursion theoretic properties, as well as noncomplemented open sets. In contrast to techniques in classical topology, we construct a disjoint recursively enumerable collection of basic open sets which cannot be extended to a recursively enumerable disjoint collection of basic open sets whose union is dense in X. ?0. Introduction. Constructions have a central role in all branches of mathematics. The question of the effectiveness of these constructions arises naturally. In this article, we employ ideas and techniques of recursion theory in order to investigate the effectiveness of certain constructions in topology. This creates an amalgamation, a framework within which other inquiries into the effective content of classical topology can be conducted, see Kalantari and Retzlaff [6]. Other areas of mathematics have undergone similar development, especially algebra. Frolich and Shepherdson [2] discussed explicit fields and their effective extensions. Rabin [16] studied computable fields and groups, their substructures and computable homomorphisms. Metakides and Nerode [12] made a major contribution to computable algebra by investigating the lattice of recursively enumerable subspaces of a fully effective vector space. A significant difference between their work and previous studies is the adaptation of the finite injury priority method to meet algebraic requirements. This innovation, and Nerode's program (for an outline see Metakides and Nerode [11]) for studying r.e. presented structures and recursively presented models has initiated much work in effective algebra. See, for example, Eisenberg [11], Remmel [11], Venning [11], Millar [11], Kalantari [3], [4], and Retzlaff [18], [19]. In this article, we begin an investigation of effective topology in a similar spirit. We study X, a fully effective topological space with a countable list z of basic open subsets in which intersection and inclusion are computable. 2(X) denotes the lattice of those open subsets of X generated by the recursively enumerable (r.e.) subsets of z, with the usual lattice operations. We begin our study of 2(X) by showing that the theory of 2(X) cannot be reduced to classical recursion theory. The new theory is rich and is subject to development using the full strength of modern recursion theory. In particular, priority arguments provide a large collection of new topological constructions. In ? 1, we list the technical and notational machinery needed to present our arguments. Furthermore, we focus on the topological and recursion theoretic properties of (X, z) and state them precisely. In ?2, we construct an open set which is not Received January 1, 1978; revised September 13, 1978. 609 ? 1979 Association for Symbolic Logic 0022-4812/79/4404-001 3/$05.25 This content downloaded from 207.46.13.162 on Fri, 01 Jul 2016 05:45:29 UTC All use subject to http://about.jstor.org/terms 610 IRAJ KALANTARI AND ALLEN RETZLAFF complemented in V(X), but which as a subset of z is recursive. This accentuates the nonreducibility of the theory of 2(X) to classical recursion theory. Next, refinements of the above construction are presented. They yield various r.e. open sets which fall into four classes with distinct recursion theoretic characterizations. ?3 contains another point of departure of effective topology from the classical theory. We construct a pairwise disjoint collection, I, of basic open sets which is not extendible to a partition of X in a strong sense. Namely, neither any set dense in I nor any set in which I is dense is extendible to a partition of X. ?1. Discussion, notation and definitions. We consider a pair (X, J) where X is a topological space and J is a countable base for the topology on X. Throughout the paper, z will satisfy the topological and recursion theoretic properties listed below. We use a, B. r' 3, a, with and without subscripts to denote elements of z (called basic open sets).

A note on $S$-closed spaces

Abstract: ABsTRAcr. A topological space X is said to be S-closed if and only if for every semi-open cover of X there exists a finite subfamily such that the union of their closures cover X. For a Hausdorff space, the concept of S-closed is shown to be equivalent to the concept of extremally disconnected and nearly compact. Further it has been shown that EDH-closed spaces are precisely S-closed Hausdorff spaces.

Some maximal topologies which are QHC

Abstract: In this paper we characterize maximal C-compact spaces, maximal QHC spaces, and maximal nearly compact spaces. We also discuss a covering property which turns out to be equivalent to S-closed and extremally disconnected. 1. C-compact spaces. In 1969 Giovanni Viglino [V1] introduced the concept of C-compact spaces and showed that a Hausdorff C-compact space is minimal Hausdorff (and thus maximal C-compact). In this first section we shall characterize all maximal C-compact spaces. The author wishes to thank the referee whose suggestions improved the exposition of this paper. The concept of a C-compact space utilizes the concept of QHC-subspaces. A topological space is quasi-H-closed (QHC) if every open cover has a finite proximate subcover; that is, every open cover has a finite subfamily whose union is dense in the space or equivalently a finite subfamily whose closures cover the space. A subset A of a topological space (X, T) is QHC if it is QHC in its relative topology. A subset A of a topological space (X, T) is QHC relative to T if every T-open cover of A has a finite subfamily whose T-closures cover the set. A QHC subset is QHC relative to T but not conversely [PT1]. A space is C-compact if every closed subset is QHC relative to T. A topological space (X, T) with property R is maximal R [minimal R] if wheneverT* :1D [T* c T], (X, T*) does not have property R. The following was proved by Viglino [Vi]: In Hausdorff spaces, compact -> C-compact -* minimal T2, and the implications are not reversible. From its very definition it is obvious that every C-compact space is QHC. Maximal QHC spaces have been studied by the author in [C2] and included in a general class of spaces in [C3]. C-compactness is not of the latter class since the properties included in this class are contractive, contagious, preserved by finite unions, semiregular, and regular closed hereditary. C-compactness is contractive [S1] and preserved by finite unions (Theorem 1), but is not contagious (Example 1), semiregular (Example 2), or regular closed hereditary [Si]. A property is contagious if whenever it is possessed by a dense subset of a space then the entire space has the property. Examples of contagious properties are connectedness, separability, and QHC. Received by the editors June 21, 1978 and, in revised form, September 6, 1978. AMS (MOS) subject classifications (1970). Primary 54D99.

Alexandroff algebras and complete regularity

Abstract: We characterize lattice theoretically the topological notion of complete regularity and study the implications of this characterization in the setting of local lattices (complete distributive lattices). In this paper, we will give a lattice theoretic characterization of complete regularity. Normality, compactness and paracompactness among others, have been considered from the point of view of lattice theory, and since they make no reference to the points of the space they can be translated almost verbatim into the language of the theory of lattices. Complete regularity is awkward on two counts: not only does it mention the points of the space explicitly, it involves directly the real numbers. However with a maneuver inspired by the work of A. D. Alexandroff we will derive a characterization which makes no reference to the points of the space or the real numbers. An analogy can be made with paracompactness-this notion is defined in a lattice theoretic manner and has a well-known characterization involving real valued functions. Frink [3] has a characterization of complete regularity which avoids mention of the real numbers but involves the points of the space directly in the characterization. Moreover his proof appeals to the existence of ultrafilters by way of a Wallman compactification. The proof of our Theorem 1.4 appeals only to a variant of Urysohn's lemma which was proved in [13]. We draw the attention of the reader to [12] which characterizes complete regularity from another point of view. In Theorem 1.6 we will show that this result can be extended to any local lattice (complete distributive lattice) which is completely regular. In ?2 we show that the familiar construction of the Baire sets can be extended to this lattice theoretic setting also. 1. Alexandroff algebras. 1.1 DEFINITION (see [8]). A local lattice L is a complete lattice which satisfies the identity x A (VY.) = V (x A Ya) for any family {Ya} C L. A homomorphism of local lattices is a function which preserves finite meets and arbitrary joins. Observe that if X and Y are topological spaces then the open subsets of X and Y form local lattices, 0 (X) and 9 (Y), respectively, and the continuous functions from X to Y are in bijective correspondence with the homomorphisms from e (Y) to e (X) provided X and Y are sober (see [9]). Received by the editors August 28, 1978 and, in revised form, December 1, 1978. AMS (MOS) subject classifications (1970). Primary 54A05; Secondary 06A35, 18F20. ? 1979 American Mathematical Society 0002-9939/79/0000-0427/$02.25

Some Topological Properties of Spaces of Holomorphic Mappings in Infinitely Many Variables

Abstract: Publisher Summary This chapter presents a study on some topological properties of spaces of holomorphic mappings in infinitely many variables. Throughout this chapter, it is assumed that E, F be complex locally convex spaces, U be a nonvoid open subset of E. There are three natural topologies that have been considered on the mentioned the vector space of all holomorphic mappings—namely, the classical compact-open topology, the topology introduced by Nachbin, and the topology introduced independently and at the same time by Coeure in the separable case and by Nachbin in the general case. The first relevant instance proving equality of two of these three topologies in infinite dimensions seems to be the result of Barroso. The chapter includes Schottenloher, Taylor series, and discusses preliminaries, several lemmas, and so on.

On countability of point-finite families of sets

Abstract: It is well known that in a separable topological space every point-finite family of open subsets is countable. In the following we are going to show that both in σ-finite measure-spaces and in topological spaces satisfying the countable chain condition, point-finite families consisting of “large” subsets are countable. Notation and terminology. Let A be a set. The family consisting of all (finite) subsets of A is denoted by . Let be a family of subsets of A. The sets and are denoted by and , respectively. We say that the family is point-finite (disjoint) if for each a ∈ A , the family has at most finitely many members (at most one member).

Subbase characterizations of compact topological spaces

Abstract: In this paper we give characterizations of some classes of compact topological spaces, such as (products of) compact lattice, tree-like and orderable spaces, by means of the existence of a closed subbase of a special kind.