On Homogeneous Images of Compact Ordered Spaces

1993 ◽  
Vol 45 (2) ◽  
pp. 380-393 ◽  
Author(s):  
J. Nikiel ◽  
E.D. Tymchatyn
Keyword(s):  
Pairwise Disjoint ◽  
Continuous Image ◽  
Closed Curves ◽  
Ordered Space ◽  
Continuous Images

AbstractWe answer a 1975 question of G R Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds(I) X is metrizable, (II) dim X = 0 or (III) X is a union of finitely many pairwise disjoint generalized simple closed curves.We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta.

scholarly journals Sum of the spaces on ordered setting

2020 ◽  
Vol 6 (2) ◽  
pp. 255-265
Author(s):  
T. M. Al-shami
Keyword(s):  
Pairwise Disjoint ◽  
Compact Spaces ◽  
Separation Axioms ◽  
Continuous Maps ◽  
Ordered Space

AbstractOne of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using open neighborhoods or neighborhoods without any difference, however, they are distinct on the ordered topology according to the neighborhoods: Are they open neighborhoods or not? In this paper, we present the concept of sum of the ordered spaces using pairwise disjoint topological ordered spaces and study main properties. Then, we introduce the properties of ordered additive, finitely ordered additive and countably ordered additive which associate topological ordered spaces with their sum. We prove that the properties of being Ti-ordered and strong Ti-ordered spaces are ordered additive, however, the properties of monotonically compact and ordered compact spaces are finitely ordered additive. Also, we define a mapping between two sums of the ordered spaces using mappings between the ordered spaces and deduce some results related to some types of continuity and homeomorphism. We complete this work by determining the conditions under which a topological ordered space is sum of the ordered spaces.

Complete systems of surfaces in 3-manifolds

1997 ◽  
Vol 122 (1) ◽  
pp. 185-191 ◽  
Author(s):  
FENGCHUN LEI
Keyword(s):  
Finite Number ◽  
Boundary Component ◽  
Pairwise Disjoint ◽  
Complete System ◽  
Closed Surface ◽  
The Other ◽  
Closed Curves ◽  
Orientable Surfaces

A complete system (CS) [Jscr ]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr ] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr ] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr ] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr ] is a CS of surfaces for M, then any CS equivalent to [Jscr ] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

Degree and holomorphic extendibility

2013 ◽  
Vol 143 (1) ◽  
pp. 129-139
Author(s):  
Darja Govekar Leban
Keyword(s):  
Continuous Function ◽  
Finite Number ◽  
Bounded Domain ◽  
Pairwise Disjoint ◽  
Continuous Functions ◽  
Closed Curves ◽  
Preceding Theorem ◽  
Holomorphic Extendibility ◽  
Class Of Functions

Recently it was shown that if D is a bounded domain in ℂ whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g ∈ A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on D which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in ℂ such that D is finitely connected and such that D is equal to the interior of D.

Separators in Continuous Images of Ordered Continua and Hereditarily Locally Connected Continua

1993 ◽  
Vol 36 (2) ◽  
pp. 154-163 ◽  
Author(s):  
J. Grispolakis ◽  
J. Nikiel ◽  
J. N. Simone ◽  
E. D. Tymchatyn
Keyword(s):  
Open Subset ◽  
Hausdorff Space ◽  
Continuous Image ◽  
Open Problems ◽  
Connected Open Subset ◽  
Locally Connected Continuum ◽  
Continuous Images

AbstractLet X be a Hausdorff space which is the continuous image of an ordered continuum. We prove that every irreducible separator of X is metrizable. This is a far reaching extension of the 1967 theorem of S. Mardešić which asserts that X has a basis of open sets with metrizable boundaries. Our first result is then used to show that, in particular, if Y is an hereditarily locally connected continuum, then for subsets of Y quasi-components coincide with components, and that the boundary of each connected open subset of Y is accessible by ordered continua. These results answer open problems in the literature due to the fourth and third authors, respectively.

scholarly journals ℵ1-directed inverse systems of continuous images of arcs

2000 ◽  
Vol 24 (2) ◽  
pp. 109-119 ◽  
Author(s):  
Ivan Lončar
Keyword(s):  
Inverse System ◽  
Continuous Image ◽  
Inverse Systems ◽  
Continuous Images

The main purpose of this paper is to prove that ifX={Xa,pab,A}is a usualℵ1-directed inverse system of continuous images of arcs with monotone bonding mappings, thenX=limXis a continuous image of an arc (Theorem 2.4). Some applications of this statement are also given.

scholarly journals Single-valued conjugates and holomorphic extendibility

2006 ◽  
Vol 136 (2) ◽  
pp. 347-350
Author(s):  
Josip Globevnik
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Complex Plane ◽  
Pairwise Disjoint ◽  
Continuous Functions ◽  
Closed Curves ◽  
Harmonic Extension ◽  
Holomorphic Extendibility

Let D be a bounded domain in the complex plane whose boundary consists of m ≥ 2 pairwise disjoint simple closed curves and let A(bD) be the algebra of all continuous functions on bD which extend holomorphically through D. We show that a continuous function Φ on bD belongs to A(bD) if for each g ∈ A (bD) the harmonic extension of Re(gΦ) to D has a single-valued conjugate.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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