In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all points that are "near" it.

In particular, in a topological space $(X,T)$, the monad of $x \in X$ is defined as $$\mu (x)=\bigcap\{{}^*O:O \in T, x \in O \}$$ (see "On Nonstandard Topology").

or the intersection of the open sets that contain $x$. For example, the monad of $0$ (given the normal topology on $\mathbb R$) is the set of infinitesimals (since any open set that contains $0$ also contains every infinitesimal).

This provides an interesting way of defining various concepts in topology:

- A set $S$ is open iff $\mu(s) \subseteq {}^*S$ for all $s \in S$ (an ideal point near a point in an open set is in the open set).
- A set $S$ is closed iff $\mu(s) \cap {}^*S \neq \emptyset$ implies $s \in S$ for all $s \in X$ (any point near an ideal point in a closed set is in that closed set).
- $(X,T)$ is compact iff for all $z \in {}^*X$, there exists $x \in X$ such that $z \in \mu (x)$ (every ideal point is near a point in a compact set).
- $(X,T)$ is hausdorff iff $\mu(x) \cap \mu (y)=\emptyset$ for all $x,y \in X, x \neq y$ (no ideal point is near two different points in a hausdorff space).
- The function $f$ from $(X,T)$ to $(Y,U)$ is continuous iff $f(\mu (x)) \subseteq \bar \mu (f (x))$ for all $x \in X$, assuming that $\mu$ is $(X,T)$'s monad function, and $\bar \mu$ is $(Y,T)$'s monad function.
- The function $f$ is a local homeomorphism iff $f(\mu (x)) = \bar \mu (f (x))$ for all $x \in X$ (and a homeomorphism if $f$ is bijective).

This has me wondering, has any one defined the concept of topology in terms of monads? The monads of a space obviously uniquely determine it, since they can be used to recover it. So, we technically could say that a topology on $X$ is a $\mu$ such that the open sets corresponding to $\mu$ satisfy the topology axioms. If we did that though, we might as well just use the regular definition! That said, there is a probably a natural definition in terms of $\mu$ making no mention of open sets (and then open sets are later a *defined* concept, instead of given).