# Reference Request: A definition of topology using monads (a.k.a. halos)

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).

• This seems very similar to the definition of a topology by specifying the filter of neighbourhoods for each point. (You just give for each point a filter containing it, then the open sets are those set that are nbds of each of their points). – Denis Nardin Feb 9 '18 at 8:28
• Oh no, another totally unrelated instance in which the term "monad" appears in mathematics! (besides category theory, and homological algebra) – Qfwfq Feb 10 '18 at 1:03
• @Qfwfq I didn't name it. – PyRulez Feb 10 '18 at 3:19
• @Qfwfq, in the current literature these are referred to as "halos" (the older term was introduced by Robinson but has been less popular than halo). – Mikhail Katz Feb 11 '18 at 14:22
• @PyRulez, notice that the concept that you mentioned, namely continuity, compactness, etc., are precisely what we need a topology to define in the traditional approach. Since continuity, compactness, etc. can now be defined without open sets, your problem has been solved as you point out! – Mikhail Katz Feb 11 '18 at 14:44

Given a nonstandard extension $X\hookrightarrow {}^\ast \!X$ (with the star operator defined for all subsets), by the first item on your list, knowledge of the halos allows you to recapture the notion of an open set. Namely, $S$ is open iff $\mu(s) \subseteq {}^*S$ for all $s \in S$. You solved your own problem then.
• In other words, given a nonstandard enlargement $X\hookrightarrow{}^\ast X$, the partition of ${}^\ast X$ into halos can itself be taken to be as the definition of "the topology", since you can recover all information about the open sets from it. – Mikhail Katz Feb 13 '18 at 11:04
• I think what the OP wants is something like the following. (1) Definition: A halo system on a (standard) set $X$ is a function $\mu$ assigning to each standard $x\in X$ a subset $\mu(x)$ of ${}^*X$ and satisfying [some list of axioms]. (2) Definition: A standard subset $A$ of $X$ is called open if $(\forall x\in A)\,\mu(x)\subseteq{}^*A$. (3) Theorem: The open sets so defined constitute a topology $T$ on $X$, and, for each $x\in X$, the halo of $x$ with respect to $T$ is the original $\mu(x)$. The crucial point will be to get the right axioms in (1). – Andreas Blass Mar 15 '18 at 0:56
• Continuation of previous comment: I think the OP intended to exclude the trivial axiomatization that just says $\mu$ should come from a topology on $X$ in the usual way. The axioms in (1) should not directly talk about topologies, so that the theorem in (3) has some content. – Andreas Blass Mar 15 '18 at 0:59