In nonstandard analysis, there is a way of studying topological spaces known as “monads”. 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)$ 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 homeomorphism iff $ f(\mu (x)) = \bar \mu (f (x))$ for all $ x$ in $ X$ .

This has me wondering, has any one defined the concept of topology in terms of monads in the literature before? 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 mine 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).