Monad (Nonstandard Analysis)

In nonstandard analysis, a monad or also a halo is the set of points infinitesimally close to a given point.^[1][2]

Given a hyperreal number x in R^∗, the monad of x is the set

${\displaystyle \text{monad}(x)=\{y\in {\mathbb{ℝ}}^{*}\mid x-y\text{ is infinitesimal}\}.}$

If x is finite (limited), the unique real number in the monad of x is called the standard part of x.^[3]

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