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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Monads of infinite points and finite product spaces


Author: Frank Wattenberg
Journal: Trans. Amer. Math. Soc. 176 (1973), 351-368
MSC: Primary 54D15; Secondary 02H25
MathSciNet review: 0312463
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Abstract: The notion of ``monad'' is generalized to infinite (i.e. non-near-standard) points in arbitrary nonstandard models of completely regular topological spaces. The behaviour of several such monad systems in finite product spaces is investigated and we prove that for paracompact spaces X such that $ X \times X$ is normal, the covering monad $ \mu $ satisfies $ \mu (x,y) = \mu (x) \times \mu (y)$ whenever x and y have the same ``order of magnitude.'' Finally, monad systems, in particular non-standard models of the real line, R, are studied and we show that in a minimal nonstandard model of R exactly one monad system exists and, in fact, $ \mu (x) = \{ x\} $ if x is infinite.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0312463-9
PII: S 0002-9947(1973)0312463-9
Keywords: Nonstandard analysis, monad systems, minimal ultrafilters, P-points
Article copyright: © Copyright 1973 American Mathematical Society