Local Topological Properties of Maps and Open Extensions of Maps

1977 ◽  
Vol 29 (6) ◽  
pp. 1121-1128
Author(s):  
J. K. Kohli
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Countable Union ◽  
Dense Open Subset ◽  
Topological Properties ◽  
Hausdorff Spaces ◽  
Local Homeomorphism ◽  
Open Map

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

scholarly journals Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.

scholarly journals On the gamma spectrum of multiplication gamma acts

2021 ◽  
Vol 48 (2) ◽  
Author(s):  
Mehdi S. Abbas ◽  
◽  
Samer A. Gubeir ◽  
Keyword(s):  
Topological Space ◽  
Gamma Spectrum ◽  
Zariski Topology ◽  
Topological Properties ◽  
Separation Axioms ◽  
Algebraic Properties

In this paper, we introduce the concept of topological gamma acts as a generalization of Zariski topology. Some topological properties of this topology are studied. Various algebraic properties of topological gamma acts have been discussed. We clarify the interplay between this topological space's properties and the algebraic properties of the gamma acts under consideration. Also, the relation between this topological space and (multiplication, cyclic) gamma act was discussed. We also study some separation axioms and the compactness of this topological space.

Separable Topological Space of Hereditary

NUTA Journal ◽  
2020 ◽  
Vol 7 (1-2) ◽  
pp. 68-70
Author(s):  
Raj Narayan Yadav ◽  
Bed Prasad Regmi ◽  
Surendra Raj Pathak
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Separable Space ◽  
Separable Topological Space ◽  
Necessary And Sufficient

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

Epimorphisms From S(X) onto S(Y)

1986 ◽  
Vol 38 (3) ◽  
pp. 538-551 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Topological Space ◽  
Hausdorff Spaces ◽  
Severe Restriction ◽  
Completely Regular ◽  
Image Position

1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T1 space X for whichforms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.

Open subset inclusion graph of a topological space

2019 ◽  
Vol 22 (6) ◽  
pp. 1007-1018
Author(s):  
Raju A. Muneshwar ◽  
Kirankumar L. Bondar
Keyword(s):  
Topological Space ◽  
Open Subset

scholarly journals Analytical Consideration of Growth in Population via Homological Invariant in Algebraic Topology

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Lewis Brew ◽  
William Obeng-Denteh ◽  
Fred Asante-Mensa
Keyword(s):  
Topological Space ◽  
Population Growth ◽  
Algebraic Topology ◽  
Homology Theory ◽  
Significant Feature ◽  
Topological Properties ◽  
Analytical Consideration ◽  
Abstract Approach ◽  
Dimensional Surface

This paper presents an abstract approach of analysing population growth in the field of algebraic topology using the tools of homology theory. For a topological space X and any point vn∈X, where vn is the n-dimensional surface, the group η=X,vn is called population of the space X. The increasing sequence from vin∈X to vjn∈X for i<j provides the bases for the population growth. A growth in population η=X,vn occurs if vin<vjn for all vin∈X and vjn∈X. This is described by the homological invariant Hηk=1. The aim of this paper is to construct the homological invariant Hηk and use Hηk=1 to analyse the growth of the population. This approach is based on topological properties such as connectivity and continuity. The paper made extensive use of homological invariant in presenting important information about the population growth. The most significant feature of this method is its simplicity in analysing population growth using only algebraic category and transformations.

scholarly journals On generalizations of C*-embedding for wallman rings

1978 ◽  
Vol 25 (2) ◽  
pp. 215-229 ◽  
Author(s):  
H. L. Bentley ◽  
B. J. Taylor
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Properties ◽  
Zero Sets ◽  
Algebraic Properties

AbstractBiles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on X in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z(A). Here we introduce two different generalizations of the concept of “a C*-embedded subset” and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

Topologies Determined by -Ideals on ω1

1978 ◽  
Vol 30 (6) ◽  
pp. 1306-1312 ◽  
Author(s):  
S. Broverman ◽  
J. Ginsburg ◽  
K. Kunen ◽  
F. D. Tall
Keyword(s):  
Measure Zero ◽  
Countable Union ◽  
Topological Properties ◽  
Natural Topology ◽  
First Category

σ-ideals (collections of sets which are closed under subset and countable union) are certainly important mathematically—consider first category sets, sets of measure zero, nonstationary sets, etc.—but aside from the observation that in certain spaces the first category σ-ideal is proper, cr-ideals have not been extensively studied by topologists. In this note we study a natural topology determined by a d-ideal, exploiting the interplay between the set-theoretic properties of the σ-ideal and the topological properties of the associated space.

On Prime Immersions of S1 into R2

1976 ◽  
Vol 28 (3) ◽  
pp. 589-593
Author(s):  
John R. Martin
Keyword(s):  
Finite Number ◽  
Open Subset ◽  
Double Point ◽  
Dense Open Subset ◽  
Double Points ◽  
Combinatorial Invariant ◽  
Linearly Independent ◽  
Oriented Plane ◽  
Intersection Sequence ◽  
Element Set

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.

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