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

Published online by Cambridge University Press:  12 March 2014

Nader Vakil
Nader Vakil*
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195
*
Department of Mathematics, Western Illinois University, Macomb, Illinois 61455
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Abstract

Let (*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 relation

Frank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of R ns to the whole of *X. Wattenberg's extensions of R ns 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 R ns to *X should be possible in any topological space. A study of such extensions of R ns 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 R ns is equivalent to the condition that the space (X, T) be regular.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 3 , September 1987 , pp. 689 - 697
Copyright
Copyright © Association for Symbolic Logic 1987

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References

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