Monad in nonstandard topology

Remarks

Example: for $X = \R$ with the usual topology, ${}^*X$ is the field of hyperreal numbers. A hyperreal $y$ lies in $\mu(x)$ exactly when $\abs{y - x} < 1/n$ for every positive integer $n$, that is, when $y - x$ is infinitesimal; so $x + \varepsilon \in \mu(x)$ for every infinitesimal $\varepsilon$. Non-example: for $x = 0$ and a positive infinitesimal $\varepsilon$, the hyperreal $1 + \varepsilon$ is not in $\mu(0)$, since it is not in ${}^*(-1/2, 1/2)$. The standard part of a finite hyperreal $y$ is the unique real $x$ with $y \in \mu(x)$. Standard characterizations in an enlargement: $X$ is Hausdorff if and only if distinct points have disjoint monads. $X$ is compact if and only if every point of ${}^*X$ lies in the monad of some point of $X$. A map $f \colon X \to Y$ is continuous at $x$ if and only if ${}^*f(\mu(x)) \subseteq \mu(f(x))$.