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]