Showing papers on "Topological space published in 1987"

Introduction to uniform spaces

Abstract: This book is based on a course taught to an audience of undergraduate and graduate students at Oxford, and can be viewed as a bridge between the study of metric spaces and general topological spaces. About half the book is devoted to relatively little-known results, much of which is published here for the first time. The author sketches a theory of uniform transformation groups, leading to the theory of uniform spaces over a base and hence to the theory of uniform covering spaces. Readers interested in general topology will find much to interest them here.

Topological homogeneity. Topological groups and their continuous images

Abstract: CONTENTSIntroduction Terminology and notation § 1. Some concepts and estimates from the theory of cardinal invariants § 2. Topological homogeneity and the exponent of a space in the Vietoris topology § 3. Homogeneity, products and retracts of homogeneous spaces § 4. Pracharacter, predensity, integrability by character, and the existence of Gδ-points in compact spaces § 5. Continuous images of topological groups § 6. Some additional results on homogeneity and topological groups References

Pointwise Compactness in Spaces of Continuous Functions

Abstract: In this paper we describe a class of topological spaces X such that Cp(X), the space of continuous functions on A'endowed with the topology of pointwise convergence, is an angelic space. This class contains the topological spaces with a dense and countably determined subspace; in particular the topological spaces which are ^-analytic in the sense of G. Choquet. Our results include previous ones of A. Grothendieck, J. L. Kelley and I. Namioka, J. D. Pryce, R. Haydon, M. De Wilde, K. Floret and M. Talagrand. As a consequence we obtain an improvement of the Eberlein-Smulian theorem in the theory of locally convex spaces. This result allows us to deduce, for instance, that (LF)-spaces and dual metric spaces, in particular (Z)F)-spaces of Grothendieck, are weakly angelic. In this way the answer to a question posed by K. Floret about the weak angelic character of (L/)-spaces is given.

Weakly α-continuous functions

Abstract: In this paper, we introduce the notion of weakly α-continuous functions in topological spaces. Weak α-continuity and subweak continuity due to Rose [1] are independent of each other and are implied by weak continuity due to Levine [2]. It is shown that weakly α-continuous surjections preserve connected spaces and that weakly α-continuous functions into regular spaces are continuous. Corollary 1 of [3] and Corollary 2 of [4] are improved as follows: If f1:X→Y is a semi continuous function into a Hausdorff space Y, f2:X→Y is either weakly α-continuous or subweakly continuous, and f1=f2 on a dense subset of X, then f1=f2 on X.

Topological structures in computer science

Abstract: Topologies of finite spaces and spaces with countably many points are investigated. It is proven, using the theory of ordered topological spaces, that any topology in connected ordered spaces, with finitely many points or in spaces similar to the set of all integers, is an interval-alternating topology. Integer and digital lines, arcs, and curves are considered. Topology of N-dimensional digital spaces is described. A digital analog of the intermediate value theorem is proven. The equivalence of connectedness and pathconnectedness in digital and integer spaces is also proven. It is shown here how methods of continuous mathematics, for example, topological methods, can be applied to objects, that used to be investigated only by methods of discrete mathematics. The significance of methods and ideas in digital image and picture processing, robotic vision, computer tomography and system's sciences presented here is well known.

Quasi F-Covers of Tychonoff Spaces

Abstract: A Tychonoff topological space is called a quasi F-space if each dense coz.ero-set of X is C*-embedded in X. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the " minimal quasi F-cover" QFf X) of a compact space X as an inverse limit space, and identify the ring C(QF(X)) as the order-Cauchy completion of the ring C*(X). In On perfect irreducible preimuges, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi F-cover of an arbitrary Tychonoff space. In this paper the minimal quasi F-cover of a compact space X is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of X. The relationship between QF(X) and QF(,BX) is studied in detail, and broad conditions under which ,B(QF(X))= QF(,BX) are obtained, together with examples of spaces for which the relationship fails. (Here ,BX denotes

Universal Loeb-measurability of sets and of the standard part map with applications

Abstract: It is shown in this paper that for K-saturated models many important external sets of nonstandard analysis-such as monadic sets or the set of ail near-standard points or all pre-near-standard points or all compact points-are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this . . dlrectlon. Moreover, the standard part map can be used as a measure preserving transformation for all -smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of -smooth Baire-measures to T-smooth Borel-measures, the decomposition theorems for -smooth Baire-measures and -smooth Borel-measures and Kakutanis theorem for product measures.

Properties of some weak forms of continuity.

Abstract: As weak forms of continuity in topological spaces, weak continuity (I), quasi continuity (2), semi continuity (3) and almost continuity in the sense of Husain (4) are well-known. Recently, the following four weak forms of continuity have been introduced: weak quasi continuity (5), faint continuity (6), subweak continuity (7) and almost weak continuity (8). These four weak forms of continuity are all weaker than weak continuity. In this paper we show that these four forms of continuity are respectively independent and investigate many fundamental properties of these four weak forms of continuity by comparing those of weak continuity, semi continuity and almost continuity.

On the congruence lattice of a frame.

Abstract: Recall that the Skula modification SkX of a topological space X is the space with the same underlying set as X whose topology is generated by the topology ΩX of X and the closed subsets of X. R. E. Hoffmann characterizes the spaces X for which SkX is compact Hausdorff as the noetherian sober spaces. The object of this note is to give a simple proof of the analogue of this characterization for frames and to show how our result for frames applies to the original one for spaces.

Topological spaces with dense subspaces that are homeomorphic to complete metric spaces and the classification of C(K) Banach spaces

Abstract: In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

Weakly Closed Functions and Hausdorff Spaces

Abstract: For weakly closed surjections between arbitrary topological spaces conditions are sought which assure a HAUSDORFF range. Some results of T. NOIRI are corrected and/or improved, and some results of J. K. KOHLI are slightly strengthened.

Properties of properly embedded minimal surfaces of finite topology

Abstract: Until the recent discovery of a sequence of properly embedded minimal surfaces with finite topology (Hoffman [4, 5]; Hoffman and Meeks [6, 7]), the only known examples were the plane, the catenoid and the helicoid. The existence of these new examples, which we will call Mk, k > 1, and others we have found (Callahan, Hoffman and Meeks [1]) makes it natural to ask qualitative questions about their behavior. It is a fundamental fact, due to Osserman [12], that if a complete minimal surface has finite total curvature, then it is, conformally, a closed Riemann surface punctured in a finite number of points; finite total curvature implies finite topology. The minimal surface Mk has total curvature equal to —47T(A; + 2). It is conformally a closed surface of genus k punctured in three points. A natural question to ask is whether or not each of the surfaces Mk lies in a one-parameter family of complete embedded minimal surfaces of finite total curvature. It is known that the plane and the catenoid are the unique embedded examples of finite total curvature with their respective topologies. In particular they cannot be perturbed through embedded examples. However

Nonlinear integral inequalities for functions defined in partially ordered topological spaces

Abstract: On considere des inegalites integrales pour des fonctions reelles definies dans des espaces topologiques partiellement ordonnes avec une mesure, les operateurs integraux sont non lineaires et essentiellement de type Volterra

Total paracompactness of real GO-spaces

Abstract: A topological space is said to be totally paracompact (resp. totally metacompact) if every open base of it has a locally finite (resp. pointfinite) subcover. In this paper we characterize all totally paracompact GOspaces constructed on the real line. It turns out that in the class of GO-spaces on the real line, total paracompactness and total metacompactness are equivalent. Another consequence of our characterization is that totally metacompact GO-spaces on the real line are metrizable. Questions and partial results are given concerning total paracompactness in subspaces of real GO-spaces. A topological space is said to be totally paracompact [Fo] (totally metacompact) if every base of it has a locally finite subcover (point-finite subcover). R. Telgarsky and H. Kok [TK] showed that the Michael Line [M] was not totally paracompact. In [Le] A. Lelek asked if the Sorgenfrey Line [S] was totally paracompact. This question was answered negatively by J. M. O'Farrell [OF1] using a technique that showed neither the Sorgenfrey Line nor the Michael Line is totally metacompact. Since both the Sorgenfrey Line and the Michael Line are real GO-spaces the following questions naturally arise: 1. What GO-spaces on the real line are totally metacompact or even totally paracompact? 2. Is total metacompactness equivalent to total paracompactness in real GOspaces? In general the answer to Question 2 is no, since Heath's "V-space" [H] is totally metacompact and not (totally) paracompact. In this paper it will be shown that total metacompactness and total paracompactness are equivalent in real GO-spaces. Moreover, we shall completely characterize all totally paracompact real GO-spaces in Theorem 2.3. From this characterization it follows that totally paracompact real GO-spaces are metrizable. Recall that a linearly ordered topological space (=LOTS) is a linearly ordered set X equipped with the usual open interval topology. If < is the linear order on X then a subset C of X is order convex if whenever a and b are in C such that a < b, then {x E XI a < x < b} C C. A generalized ordered space (=GO-space) is a linearly ordered set equipped with a T1-topology for which there is a base consisting of convex sets. GO-spaces have been studied extensively (for example, see [BL1] or Received by the editors December 10, 1985 and, in revised form, August 14, 1986. The results in this paper were presented at the University of Southwest Louisiana Spring Topology Conference in April of 1986. This conference was sponsored in part by N.S.F. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F05, 54D18; Secondary 54E35.