scholarly journals An extension theorem for sober spaces and the Goldman topology

2003 ◽  
Vol 2003 (51) ◽  
pp. 3217-3239 ◽  
Author(s):  
Ezzeddine Bouacida ◽  
Othman Echi ◽  
Gabriel Picavet ◽  
Ezzeddine Salhi
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Extension Theorem ◽  
Topological Characterization ◽  
Continuous Maps

Goldman points of a topological space are defined in order to extend the notion of primeG-ideals of a ring. We associate to any topological space a new topology called Goldman topology. For sober spaces, we prove an extension theorem of continuous maps. As an application, we give a topological characterization of the Jacobson subspace of the spectrum of a commutative ring. Many examples are provided to illustrate the theory.

The Minimal Prime Spectrum of a Commutative Ring

1971 ◽  
Vol 23 (5) ◽  
pp. 749-758 ◽  
Author(s):  
M. Hochster
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Topological Characterization ◽  
Completely Regular ◽  
Minimal Prime

We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

“Topologically Indexed Function Spaces and Adjoint Functors”

1982 ◽  
Vol 25 (2) ◽  
pp. 169-178
Author(s):  
S. B. Niefield
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Topological Spaces ◽  
Product Topology ◽  
Adjoint Functors ◽  
Point Space ◽  
Continuous Maps ◽  
Necessary And Sufficient

AbstractLet Top denote the category of topological spaces and continuous maps. In this paper we discuss families of function spaces indexed by the elements of a topological space T, and their relationship to the characterization of right adjoints Top/S → Top/T, where S is also a topological space. After reducing the problem to the case where S is a one-point space, we describe a class of right adjoints Top → Top/T, and then show that every right adjoint Top → Top/T is isomorphic to one of this form. We conclude by giving necessary and sufficient conditions for a left adjoint Top/T → Top to be isomorphic to one of the form − XTY, where Y is a space over T, and xT denotes the fiber product with the product topology.

scholarly journals Homotopy Characterization of ANR Function Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jaka Smrekar
Keyword(s):  
Topological Space ◽  
Homotopy Type ◽  
Metric Spaces ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Cw Complex ◽  
Continuous Maps ◽  
Compactly Generated

LetYbe an absolute neighbourhood retract (ANR) for the class of metric spaces and letXbe a topological space. LetYXdenote the space of continuous maps fromXtoYequipped with the compact open topology. We show that ifXis a compactly generated Tychonoff space andYis not discrete, thenYXis an ANR for metric spaces if and only ifXis hemicompact andYXhas the homotopy type of a CW complex.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

Intersections of quotient rings and Prüfer v-multiplication domains

2016 ◽  
Vol 15 (08) ◽  
pp. 1650149 ◽  
Author(s):  
Said El Baghdadi ◽  
Marco Fontana ◽  
Muhammad Zafrullah
Keyword(s):  
Integral Domain ◽  
Algebraic Approach ◽  
Quotient Field ◽  
Topological Characterization ◽  
Locally Finite ◽  
Star Operation ◽  
Star Operations ◽  
Quotient Rings ◽  
Special Case

Let [Formula: see text] be an integral domain with quotient field [Formula: see text]. Call an overring [Formula: see text] of [Formula: see text] a subring of [Formula: see text] containing [Formula: see text] as a subring. A family [Formula: see text] of overrings of [Formula: see text] is called a defining family of [Formula: see text], if [Formula: see text]. Call an overring [Formula: see text] a sublocalization of [Formula: see text], if [Formula: see text] has a defining family consisting of rings of fractions of [Formula: see text]. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations [D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988) 2535–2553]. We show as a consequence of our work that domains that are locally finite intersections of Prüfer [Formula: see text]-multiplication (respectively, Mori) sublocalizations turn out to be Prüfer [Formula: see text]-multiplication domains (PvMDs) (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of Théorème 1 of [J. Querré, Intersections d’anneaux intègers, J. Algebra 43 (1976) 55–60] and Proposition 3.2 of [N. Dessagnes, Intersections d’anneaux de Mori — exemples, Port. Math. 44 (1987) 379–392]. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing P vMDs as a subclass of the class of essential domains (see also Theorem 2.4 of [C. A. Finocchiaro and F. Tartarone, On a topological characterization of Prüfer [Formula: see text]-multiplication domains among essential domains, preprint (2014), arXiv:1410.4037]).

Topological characterization of dendrimer, benzenoid, and nanocone

2018 ◽  
Vol 74 (1-2) ◽  
pp. 35-43
Author(s):  
Wei Gao ◽  
Muhammad Kamran Siddiqui ◽  
Najma Abdul Rehman ◽  
Mehwish Hussain Muhammad
Keyword(s):  
Chemical Properties ◽  
Chemical Compounds ◽  
Topological Indices ◽  
Zagreb Index ◽  
Topological Characterization ◽  
Complex Molecules ◽  
Chemical Structures ◽  
Physico Chemical ◽  
Quantitative Structure Activity Relationships

Abstract Dendrimers are large and complex molecules with very well defined chemical structures. More importantly, dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core. Topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure-activity relationships. These topological indices correlate certain physico-chemical properties such as the boiling point, stability, strain energy, and others, of chemical compounds. In this article, we determine hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for hetrofunctional dendrimers, triangular benzenoids, and nanocones.

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