Concurrent Relation
المؤلف:
Hurd, A. E. and Loeb, P. A
المصدر:
An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.
الجزء والصفحة:
...
9-1-2022
1285
Concurrent Relation
Let 
and 
be sets, and let 
be a relation on 
. Then 
is a concurrent relation if and only if for any finite subset 
of 
, there exists a single element 
of 
such that if 
, then 
. Examples of concurrent relations include the following:
1. The relation 
on either the natural numbers, the integers, the rational numbers, or the real numbers.
2. The relation 
between elements of an extension 
of a field 
, defined by
![R=<span style=]() {(a,b) in E×E:b is algebraic over F and
a is in the extension of F by b}. " src="https://mathworld.wolfram.com/images/equations/ConcurrentRelation/NumberedEquation1.gif" style="height:37px; width:263px" /> |
3. The containment relation 
between open neighborhoods of a given point 
of a topological space 
.
REFERENCES:
Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.
Robinson, A. "Germs." In Applications of Model Theory to Algebra, Analysis and Probability (International Sympos., Pasadena, Calif., 1967). New York: Holt, Rinehart and Winston, pp. 138-149, 1969.
Insall, M. "Hyperalgebraic Primitive Elements for Relational Algebraic and Topological Algebraic Models." Studia Logica 57, 409-418, 1996.
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