Monads of infinite points and finite product spaces

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.