Monadic binary relations and the monad systems at near-standard points

1987 ◽  
Vol 52 (3) ◽  
pp. 689-697
Author(s):  
Nader Vakil
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Hausdorff Space ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Completely Regular ◽  
General Topological Space ◽  
Image Position

AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

scholarly journals Multiplication operators on weighted spaces in the non-locally convex framework

1997 ◽  
Vol 20 (1) ◽  
pp. 75-79 ◽  
Author(s):  
L. A. Khan ◽  
A. B. Thaheem
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Weighted Space ◽  
Topological Algebra ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Completely Regular ◽  
General Topological Space ◽  
Necessary And Sufficient ◽  
Locally Convex

LetXbe a completely regular Hausdorff space,Ea topological vector space,Va Nachbin family of weights onX, andCV0(X,E)the weighted space of continuousE-valued functions onX. Letθ:X→Cbe a mapping,f∈CV0(X,E)and defineMθ(f)=θf(pointwise). In caseEis a topological algebra,ψ:X→Eis a mapping then defineMψ(f)=ψf(pointwise). The main purpose of this paper is to give necessary and sufficient conditions forMθandMψto be the multiplication operators onCV0(X,E)whereEis a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption thatEis locally convex.

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.

scholarly journals Relations on topological spaces: Urysohn's lemma

1968 ◽  
Vol 8 (1) ◽  
pp. 37-42
Author(s):  
Y.-F. Lin
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Order Relation ◽  
Topological Spaces ◽  
Topological Closure ◽  
Image Position

Let X be a topological space equipped with a binary relation R; that is, R is a subset of the Cartesian square X×X. Following Wallace [5], we write Deviating from [7], we shall follow Wallace [4] to call the relation R continuous if RA*⊂(RA)* for each A⊂X, where * designates the topological closure. Borrowing the language from the Ordered System, though our relation R need not be any kind of order relation, we say that a subset S of X is R-decreasing (R-increasing) if RS ⊂ S(SR ⊂ S), and that S is Rmonotone if S is either R-decreasing or R-increasing. Two R-monotone subsets are of the same type if they are either both R-decresaing or both Rincreasing.

Variable Neighborhood Model for Agent Control Introducing Accessibility Relations Between Agents with Linear Temporal Logic

2014 ◽  
Vol 18 (6) ◽  
pp. 937-945
Author(s):  
Seiki Ubukata ◽  
◽  
Tetsuya Murai ◽  
Yasuo Kudo ◽  
Seiki Akama ◽  
...  
Keyword(s):  
Topological Space ◽  
Temporal Logic ◽  
Binary Relation ◽  
Linear Temporal Logic ◽  
Binary Relations ◽  
Cellular Space ◽  
Current Time ◽  
Low Level ◽  
Local Properties ◽  
Agent Control

In general, there are two types of agents, reflex and deliberative. The former does not have the ability for deep planning that produces higher-level actions to attain goals cooperatively, which is the ability of the latter. Can we cause reflex agents to act as though they could plan their actions? In this paper, we propose a variable neighborhood model for reflex agent control, that allows such agents to create plans in order to attain their goals. The model consists of three layers: (1) topological space, (2) agent space, and (3) linear temporal logic. Agents with their neighborhoods move in a topological space, such as a plane, and in a cellular space. Then, a binary relation between agents is generated each time from the agents’ position and neighborhood. We call the pair composed of a set of agents and binary relations the agent space. In order to cause reflex agents to have the ability to attain goals superficially, we consider the local properties of the binary relation between agents. For example, if two agents have a symmetrical relation at the current time, they can struggle to maintain symmetry or they could abandon symmetry at the next time, depending on the context. Then, low-level behavior, that is, the maintenance or abandonment of the local properties of binary relations, grant reflex agents a method for selecting neighborhoods for the next time. As a result, such a sequence of low-level behavior generates seemingly higher-level actions, as though reflex agents could attain a goal with such actions. This low-level behavior is shown through simulation to generate the achievement of a given goal, such as cooperation and target pursuing.

scholarly journals On weak approximation and convexification in weighted spaces of vector-valued continuous functions

1989 ◽  
Vol 31 (1) ◽  
pp. 59-64 ◽  
Author(s):  
Marek Nawrocki
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Weak Approximation ◽  
Completely Regular ◽  
Hausdorff Topological Vector Space ◽  
The Family ◽  
Image Position ◽  
Positive Functions ◽  
Vector Valued

Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the formwhere υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.

Separating Closed Sets by Continuous Mappings into Developable Spaces

1981 ◽  
Vol 33 (6) ◽  
pp. 1420-1431 ◽  
Author(s):  
Harald Brandenburg
Keyword(s):  
Topological Space ◽  
Double Sequence ◽  
Completely Regular ◽  
Closed Sets ◽  
Continuous Mappings ◽  
Neighbourhood Base ◽  
Image Position ◽  
Metrizable Spaces ◽  
Metrization Theorem ◽  
Developable Spaces

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, whereThis class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

Approximations in Rough Sets vs Granular Computing for Coverings

2012 ◽  
pp. 152-163
Author(s):  
Guilong Liu ◽  
William Zhu
Keyword(s):  
Knowledge Discovery ◽  
Binary Relation ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Knowledge Discovery In Databases ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Covering Rough Set

Rough set theory is an important technique in knowledge discovery in databases. Classical rough set theory proposed by Pawlak is based on equivalence relations, but many interesting and meaningful extensions have been made based on binary relations and coverings, respectively. This paper makes a comparison between covering rough sets and rough sets based on binary relations. This paper also focuses on the authors’ study of the condition under which the covering rough set can be generated by a binary relation and the binary relation based rough set can be generated by a covering.

scholarly journals 0-dimensional compactifications and Boolean rings

1968 ◽  
Vol 8 (4) ◽  
pp. 755-765 ◽  
Author(s):  
K. D. Magill ◽  
J. A. Glasenapp
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Hausdorff Space ◽  
Dimensional Space ◽  
Continuous Mapping ◽  
Dense Subspace ◽  
Clopen Subset ◽  
Partially Order ◽  
Completely Regular ◽  
Boolean Rings

A subset of a topological space which is both closed and open is referred to as a clopen subset. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. By a compactification αX of a completely regular Hausdorff space X, we mean any compact space which contains X as a dense subspace. Two compactifications αX and γX are regarded as being equivalent if there exists a homeomorphism from αX onto γX which keeps X pointwise fixed. We will not distinguish between equivalent compactifications. With this convention, we can partially order any family of compactifications of X by defining αX ≧ γX if there exists a continuous mapping from γX onto αX which leaves X pointwise fixed. This paper is concerned with the study of the partially ordered family [X] of all 0-dimensional compactifications of a 0-dimensional space X.

Approximations in Rough Sets vs Granular Computing for Coverings

2010 ◽  
Vol 4 (2) ◽  
pp. 63-76 ◽  
Author(s):  
Guilong Liu ◽  
William Zhu
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Knowledge Discovery In Databases ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Covering Rough Set

Rough set theory is an important technique in knowledge discovery in databases. Classical rough set theory proposed by Pawlak is based on equivalence relations, but many interesting and meaningful extensions have been made based on binary relations and coverings, respectively. This paper makes a comparison between covering rough sets and rough sets based on binary relations. This paper also focuses on the authors’ study of the condition under which the covering rough set can be generated by a binary relation and the binary relation based rough set can be generated by a covering.

scholarly journals COMPLEXITY OF EQUIVALENCE RELATIONS AND PREORDERS FROM COMPUTABILITY THEORY

2014 ◽  
Vol 79 (3) ◽  
pp. 859-881 ◽  
Author(s):  
EGOR IANOVSKI ◽  
RUSSELL MILLER ◽  
KENG MENG NG ◽  
ANDRÉ NIES
Keyword(s):  
Complexity Theory ◽  
Polynomial Time ◽  
Equivalence Relation ◽  
Computability Theory ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Exponential Time ◽  
Relative Complexity ◽  
Turing Reducibility ◽  
Image Position

AbstractWe study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relationsR,S, a componentwise reducibility is defined byR≤S⇔ ∃f∀x, y[x R y↔f(x)S f(y)].Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a${\rm{\Pi }}_1^0$-complete equivalence relation, but no${\rm{\Pi }}_k^0$-complete fork≥ 2. We show that${\rm{\Sigma }}_k^0$preorders arising naturally in the above-mentioned areas are${\rm{\Sigma }}_k^0$-complete. This includes polynomial timem-reducibility on exponential time sets, which is${\rm{\Sigma }}_2^0$, almost inclusion on r.e. sets, which is${\rm{\Sigma }}_3^0$, and Turing reducibility on r.e. sets, which is${\rm{\Sigma }}_4^0$.

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