Monad in nonstandard topology

Summary

Let $X$ be a topological space and ${}^*X$ its extension in nonstandard analysis. The monad of a point $x$ of $X$ is the set of points of ${}^*X$ that are infinitely close to $x$. It is the intersection of the nonstandard extensions of all open neighbourhoods of $x$. Monads turn topological notions into simple statements about infinitely close points. For example, a space is Hausdorff exactly when distinct points have disjoint monads.