scholarly journals Elementary topology and universal computation

2008 ◽  
Vol 27 (4) ◽  
pp. 287-293
Author(s):  
Petrus Potgieter
Keyword(s):  
Topological Space ◽  
General Framework ◽  
Natural Numbers ◽  
Universal Computation ◽  
Arbitrary Topological Space ◽  
Elementary Topology

This paper attempts to define a general framework for computability on an arbitrary topological space X . The elements of X are taken as primitives in this approach—also for the coding of functions — and, except when X = N, the natural numbers are not used directly.

A General Framework for Priority Arguments

1995 ◽  
Vol 1 (2) ◽  
pp. 189-201 ◽  
Author(s):  
Steffen Lempp ◽  
Manuel Lerman
Keyword(s):  
Information Content ◽  
Equivalence Relation ◽  
Decision Problem ◽  
General Framework ◽  
Equivalence Classes ◽  
Partial Ordering ◽  
Decision Problems ◽  
Natural Numbers ◽  
Relative Complexity ◽  
Degrees Of Unsolvability

The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post [7] as an attempt to measure theinformation contentof sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in particular, they wished to know when a solution to one decision problem contained the information necessary to solve a second decision problem. As decision problems can be coded by sets of natural numbers, this question is equivalent to: Given a computer with access to an oracle which will answer membership questions about a setA, can a program (allowing questions to the oracle) be written which will correctly compute the answers to all membership questions about a setB? If the answer is yes, then we say thatBisTuring reducibletoAand writeB≤TA. We say thatB≡TAifB≤TAandA≤TB. ≡Tis an equivalence relation, and ≤Tinduces a partial ordering on the corresponding equivalence classes; the poset obtained in this way is called thedegrees of unsolvability, and elements of this poset are calleddegrees.Post was particularly interested in computability from sets which are partially generated by a computer, namely, those for which the elements of the set can be enumerated by a computer.

Constructing ω-stable structures: rank 2 fields

2000 ◽  
Vol 65 (1) ◽  
pp. 371-391 ◽  
Author(s):  
John T. Baldwin ◽  
Kitty Holland
Keyword(s):  
General Framework ◽  
Saturated Model ◽  
Closed Field ◽  
Morley Rank ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Fast Growing ◽  
Rank 2 ◽  
Stable Structures

AbstractWe provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from ‘primitive extensions’ to the natural numbers a theory Tμ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable.

Classifying positive equivalence relations

1983 ◽  
Vol 48 (3) ◽  
pp. 529-538 ◽  
Author(s):  
Claudio Bernardi ◽  
Andrea Sorbi
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Point Of View ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Natural Numbers ◽  
Uniformity Property

AbstractGiven two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, y ∈ ω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

scholarly journals EXTENSION OF CONTINUOUS MAPPINGS AND H1-RETRACTS

2008 ◽  
Vol 78 (3) ◽  
pp. 497-506 ◽  
Author(s):  
OLENA KARLOVA
Keyword(s):  
Topological Space ◽  
Continuous Mapping ◽  
Paracompact Space ◽  
Open Set ◽  
Continuous Mappings ◽  
Arbitrary Topological Space ◽  
Completely Metrizable

AbstractWe prove that any continuous mapping f:E→Y on a completely metrizable subspace E of a perfect paracompact space X can be extended to a Lebesgue class one mapping g:X→Y (that is, for every open set V in Y the preimage g−1(V ) is an Fσ-set in X) with values in an arbitrary topological space Y.

scholarly journals UNIFORM EXHAUSTIVITY OF A FAMILY OF REGULAR SET FUNCTIONS ON TOPOLOGICAL SPACES

2017 ◽  
Vol 18 (9) ◽  
pp. 61-69
Author(s):  
T.A. Sribnaya
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Set Functions ◽  
Arbitrary Topological Space

Conditions for the uniform exhaustivity of a family of regular set functions defined on an algebra £ of subsets of a cr-topological space and taking values in arbitrary topological space are found.

scholarly journals Intrinsic characterizations of C-realcompact spaces

2021 ◽  
Vol 22 (2) ◽  
pp. 295
Author(s):  
Sudip Kumar Acharyya ◽  
Rakesh Bharati ◽  
Atasi Deb Ray
Keyword(s):  
Topological Space ◽  
Dense Subspace ◽  
Finite Intersection ◽  
Compact Spaces ◽  
Closed Sets ◽  
Finite Intersection Property ◽  
Arbitrary Topological Space ◽  
Stable Family ◽  
Definition Of

<pre>c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that  X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.</pre>

Singular Integrals are Perron Integrals of a Certain Type

1970 ◽  
Vol 22 (2) ◽  
pp. 260-264
Author(s):  
W. F. Pfeffer
Keyword(s):  
Topological Space ◽  
Singular Integrals ◽  
Convergence Theorems ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Basic Properties ◽  
Arbitrary Topological Space ◽  
Principal Value

In [7] a Perron-like integral was denned in an arbitrary topological space and many of its basic properties were established. In this paper we shall show (the theorem in § 2) that in a suitable setting the integral from [7] includes a class of so-called singular integrals, i.e., generalized forms of the Cauchy principal value of an integral. Thus, the powerful machinery of Perron integration, e.g., the monotone and dominant convergence theorems, can be automatically applied to these singular integrals.

A Note on Certain Spaces with Bases (mod K)

1975 ◽  
Vol 27 (2) ◽  
pp. 469-474
Author(s):  
Harold R. Bennett ◽  
Harold W. Martin
Keyword(s):  
Topological Space ◽  
Open Covering ◽  
Compact Sets ◽  
Natural Numbers ◽  
Open Set

In this note all spaces are assumed to be regular T1 spaces and all undefined terms and notations may be found in [8], In particular let cl(A) denote the closure of the set A and let Z+ denote the set of natural numbers.Definition 1. Let X be a topological space and a covering of X by compact sets. An open covering of X is said to be a basis (mod K) if whenever and an open set V contains Kx, then there exists such that . In such a case X is written as the ordered triple .

Methodological Reflections on Typologies for Numerical Notations

2012 ◽  
Vol 25 (2) ◽  
pp. 155-195 ◽  
Author(s):  
Theodore Reed Widom ◽  
Dirk Schlimm
Keyword(s):  
Crucial Role ◽  
General Framework ◽  
World Wide ◽  
Cultural Development ◽  
Natural Numbers ◽  
Starting Point ◽  
Refined Classification

ArgumentPast and present societies world-wide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. The present paper provides a general framework for assessing the efficacy of these typologies relative to certain desiderata, and it uses this framework to discuss the two influential typologies of Zhang & Norman and Chrisomalis. Following this, a new typology is presented that takes as its starting point the principles by which numerical notations represent multipliers (the principles of cumulation and cipherization), and bases (those of integration, parsing, and positionality). Many different examples show that this new typology provides a more refined classification of numerical notations than the ones put forward previously. In addition, the framework provided here can be used to assess typologies not only of numerical notations, but also of many other domains.

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