Monad (nonstandard analysis)

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

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

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

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