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.

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.

Controlling the dependence degree of a recursively enumerable vector space

1978 ◽  
Vol 43 (1) ◽  
pp. 13-22 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Vector Space ◽  
Topological Structure ◽  
Independent Set ◽  
Original Data ◽  
Recursion Theory ◽  
Vector Spaces ◽  
Additional Structure ◽  
Recursively Enumerable ◽  
Added Benefit ◽  
And Algebra

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.

G. Metakides and A. Nerode. Recursion theory and algebra. Algebra and logic, Papers from the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia, edited by J. N. Crossley, Lecture notes in mathematics, vol. 450, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 209–219. - Iraj Kalantari and Allen Retzlaff. Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces. The journal of symbolic logic, vol. 42 no. 4 (for 1977, pub. 1978), pp. 481–491. - Iraj Kalantari. Major subspaces of recursively enumerable vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 293–303. - J. Remmel. A r-maximal vector space not contained in any maximal vector space. The journal of symbolic logic, vol. 43 (1978), pp. 430–441. - Allen Retzlaff. Simple and hyperhypersimple vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 260–269. - J. B. Remmel. Maximal and cohesive vector spaces. The journal of symbolic logic, vol. 42 no. 3 (for 1977, pub. 1978), pp. 400–418. - J. Remmel. On r.e. and co-r.e. vector spaces with nonextendible bases. The journal of symbolic logic, vol. 45 (1980), pp. 20–34. - M. Lerman and J. B. Remmel. The universal splitting property: I. Logic Colloquim '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 181–207. - J. B. Remmel. Recursively enumerable Boolean algebras. Annals of mathematical logic, vol. 15 (1978), pp. 75–107. - J. B. Remmel. r-Maximal Boolean algebras. The journal of symbolic logic, vol. 44 (1979), pp. 533–548. - J. B. Remmel. Recursion theory on algebraic structures with independent sets. Annals of mathematical logic, vol. 18 (1980), pp. 153–191. - G. Metakides and J. B. Remmel. Recursion theory on orderings. I. A model theoretic setting. The journal of symbolic logic, vol. 44 (1979), pp. 383–402. - J. B. Remmel. Recursion theory on orderings. II. The journal of symbolic logic, vol. 45 (1980), pp. 317–333.

1986 ◽  
Vol 51 (1) ◽  
pp. 229-232
Author(s):  
Henry A. Kierstead
Keyword(s):  
New York ◽  
Mathematical Logic ◽  
Vector Space ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Symbolic Logic ◽  
Recursively Enumerable ◽  
North Holland Publishing ◽  
And Algebra

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 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.

Degrees of recursively enumerable topological spaces

1983 ◽  
Vol 48 (3) ◽  
pp. 610-622 ◽  
Author(s):  
Iraj Kalantari ◽  
J. B. Remmel
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
The Body ◽  
Topological Spaces ◽  
Open Set ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Nowhere Dense Set ◽  
Dense Set

In [5], Metakides and Nerode introduced the study of recursively enumerable (r.e.) substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff consideredX, a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets ofωgive rise to r.e. subsets ofX. The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset ofX? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above.In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that givenAany r.e. set andany r.e. open subset ofX, there exists an r.e. open set ℋ which is a subset ofand is dense in(in a topological sense) and in whichAis coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree ofA. We then go on and establish various results (both existential and universal) on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2.

Recursion theory in a lower semilattice

1992 ◽  
Vol 57 (3) ◽  
pp. 892-911 ◽  
Author(s):  
Alex Feldman
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Algebraic Structure ◽  
Algebraic Closure ◽  
Recursion Theory ◽  
Vector Spaces ◽  
Algebraic Structures ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Algebraically Closed Fields

In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.The idea of looking at the lattice of recursively enumerable substructures of some recursive algebraic structure was introduced by Metakides and Nerode in [5], and since then many different kinds of algebraic structures have been studied in this way, including vector spaces, Boolean algebras, groups, algebraically closed fields, and equivalence relations. Since different algebraic structures have different recursion theoretic properties, one natural question is whether an algebraic structure with relatively little structure (such as a partial order or an equivalence relation) exhibits behavior more like classical recursion theory than one with more structure (such as vector spaces or algebraically closed fields).In [6] and [7], Metakides and Remmel studied recursion theory on orderings, and, as they point out in [6], orderings differ from most other algebraic structures in that the algebraic closure operation on orderings is trivial; but this does not present a problem for them, given the questions they explore. Moreover, they take an approach of proving general theorems which can then be applied to specific orderings. Our tack is different, although also well-established (see, for example, [3]), in which a “largest” structure is defined (in §3) which corresponds to the natural numbers in classical recursion theory. In order to distinguish substructures from subsets, a function symbol is added, namely greatest lower bound. The greatest lower bound function is fundamental to the study of orderings and occurs naturally in many of them, and thus is an appropriate addition to the theory of orderings. In §4 we redefine the concepts of simple and maximal in a manner appropriate to this structure, and prove several existence theorems.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

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