Maximality in effective topology

1983 ◽  
Vol 48 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Iraj Kalantari ◽  
Anne Leggett
Keyword(s):  
Topological Space ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Mathematical Structures ◽  
Topological Vector Spaces ◽  
Linear Orderings ◽  
Recursively Enumerable ◽  
Countable Basis ◽  
Effective Analysis

In this paper we continue the study of the structure of the lattice of recursively enumerable (r.e.) open subsets of a topological space. Work in this approach to effective topology began in Kalantari and Retzlaff [5] and continued in Kalantari [2], Kalantari and Leggett [3] and Kalantari and Remmel [4]. Studies in effectiveness of results in structures other than integers began with the work of Specker [17] and Lacombe [8] on effective analysis.The renewed activity in the study of the effective content of mathematical structures owes much to Nerode's program and Metakides' and Nerode's [11], [12] work on vector spaces and fields. These studies have been extended by Kalantari, Remmel, Retzlaff, Shore and Smith. Similar studies on the effective content of other mathematical structures have been conducted. These include work on topological vector spaces, boolean algebras, linear orderings etc.Kalantari and Retzlaff [5] began a study of effective topological spaces by considering a topological space with a countable basis ⊿ for the topology. The space X is to be fully effective; that is, the basis elements are coded into ω and the operations of intersection of basis elements and the relation of inclusion among them are both computable. An r.e. open subset of X is then represented as the union of basic open sets whose codes lie in an r.e. subset of ω.

Simplicity in effective topology

1982 ◽  
Vol 47 (1) ◽  
pp. 169-183 ◽  
Author(s):  
Iraj Kalantari ◽  
Anne Leggett
Keyword(s):  
Topological Space ◽  
Recursion Theory ◽  
Activity Analysis ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Mathematical Structures ◽  
Recursively Enumerable ◽  
Countable Basis ◽  
Recursive Analysis ◽  
And Algebra

The recursion-theoretic study of mathematical structures other thanωis now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others.Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological spaceXwith a countable basis ⊿ for the topology. The space isfully effective, that is, the basis elements are coded intoωand the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to, the lattice of recursively enumerable (r.e.) subsets ofω, the collection of r.e. open subsets ofXforms a latticeℒ(X)under the usual operations of union and intersection.

scholarly journals Tilings in topological spaces

1999 ◽  
Vol 22 (3) ◽  
pp. 611-616 ◽  
Author(s):  
F. G. Arenas
Keyword(s):  
Topological Space ◽  
Pairwise Disjoint ◽  
The Other ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
Other Hand ◽  
Extensive Bibliography

Atilingof a topological spaceXis a covering ofXby sets (calledtiles) which are the closures of their pairwise-disjoint interiors. Tilings ofℝ2have received considerable attention (see [2] for a wealth of interesting examples and results as well as an extensive bibliography). On the other hand, the study of tilings of general topological spaces is just beginning (see [1, 3, 4, 6]). We give some generalizations for topological spaces of some results known for certain classes of tilings of topological vector spaces.

scholarly journals Newel-pseudotopologiese vektorruimtes

1999 ◽  
Vol 18 (3) ◽  
pp. 89-93
Author(s):  
M. A. Muller
Keyword(s):  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces

Pseudo-topological spaces (i.e. limit spaces) were defined by Fischer in 1959. In this paper the theory of fuzzy pseudo-topological spaces is applied to vector spaces. We introduce the concept of boundedness in fuzzy pseudo-topological vector spaces.

scholarly journals M – STAR – Irresolute Topological Vector Spaces

2021 ◽  
Vol 7 ◽  
pp. 20-36
Author(s):  
Raja Mohammad Latif
Keyword(s):  
Vector Space ◽  
Extreme Point ◽  
Topological Vector Space ◽  
Convex Subset ◽  
Hausdorff Space ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
New Class

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

Recursion theory on orderings. I. A model theoretic setting

1979 ◽  
Vol 44 (3) ◽  
pp. 383-402 ◽  
Author(s):  
G. Metakides ◽  
J.B. Remmel
Keyword(s):  
Algebraic Closure ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Formal Definition ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Partial Orderings ◽  
Definition Of ◽  
Algebraically Closed Fields

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a setS, denoted cl(S), is given in §1, however in vector spaces, cl(S) is just the subspace generated byS, in Boolean algebras, cl(S) is just the subalgebra generated byS, and in algebraically closed fields, cl(S) is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl(S) = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings.

Recursive constructions in topological spaces

1979 ◽  
Vol 44 (4) ◽  
pp. 609-625 ◽  
Author(s):  
Iraj Kalantari ◽  
Allen Retzlaff
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Recursively Enumerable ◽  
Recursive Constructions

AbstractWe study topological constructions in the recursion theoretic framework of the lattice of recursively enumerable open subsets of a topological spaceX. 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 inX.

scholarly journals Internal functionals and bundle duals

1984 ◽  
Vol 7 (4) ◽  
pp. 689-695 ◽  
Author(s):  
Joseph W. Kitchen ◽  
David A. Robbins
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
The Given ◽  
Locally Convex

Ifπ:E→Xis a bundle of Banach spaces,Xcompact Hausdorff, a fibered spaceπ*:E*→Xcan be constructed whose stalks are the duals of the stalks of the given bundle and whose sections can be identified with the “functionals” studied by Seda in [1] and [2] or elements of the “internal dual”Mod(Γ(π),C(X))studied by Gierz in [3]. If the given bundle is separable and norm continuous, then the fibered spaceπ*:E*→Xis actually a full bundle of locally convex topological vector spaces (Theorem 3). In the second portion of the paper two results are stated, both of them corollaries of theorems by Gierz, concerning functionals for bundles of Banach spaces which arise, in turn, from “fields of topological spaces.”

scholarly journals Bornologiese pseudotopologiese vektorruimtes

1990 ◽  
Vol 9 (1) ◽  
pp. 15-18
Author(s):  
M. A. Muller
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Direct Sums ◽  
Paper Properties ◽  
Topological Vector Spaces ◽  
Quotient Spaces ◽  
Locally Convex

Homological spaces were defined by Hogbe-Nlend in 1971 and pseudo-topological spaces by Fischer in 1959. In this paper properties of bornological pseudo-topological vector spaces are investigated. A characterization of such spaces is obtained and it is shown that quotient spaces and direct sums o f boruological pseudo-topological vector spaces are bornological. Every bornological locally convex pseudo-topological vector space is shown to be the inductive limit in the category of pseudo-topological vector spaces of a family of locally convex topological vector spaces.

scholarly journals The Fuzzification of Classical Structures: A General View

2015 ◽  
Vol 10 (6) ◽  
pp. 12 ◽  
Author(s):  
Ioan Dzitac
Keyword(s):  
Metric Spaces ◽  
50Th Anniversary ◽  
Vector Spaces ◽  
Fuzzy Mathematics ◽  
Future Research ◽  
Topological Spaces ◽  
General View ◽  
Linear Spaces ◽  
Topological Vector Spaces ◽  
Fuzzy Topological Spaces

The aim of this survey article, dedicated to the 50th anniversary of Zadeh’s pioneering paper "Fuzzy Sets" (1965), is to offer a unitary view to some important spaces in fuzzy mathematics: fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces. We believe that this paper will be a support for future research in this field.

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