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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).

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    $\begingroup$ 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). $\endgroup$ – Denis Nardin Feb 9 '18 at 8:28
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    $\begingroup$ Oh no, another totally unrelated instance in which the term "monad" appears in mathematics! (besides category theory, and homological algebra) $\endgroup$ – Qfwfq Feb 10 '18 at 1:03
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    $\begingroup$ @Qfwfq I didn't name it. $\endgroup$ – PyRulez Feb 10 '18 at 3:19
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    $\begingroup$ @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). $\endgroup$ – Mikhail Katz Feb 11 '18 at 14:22
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    $\begingroup$ @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! $\endgroup$ – Mikhail Katz Feb 11 '18 at 14:44
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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.

The concept that you mentioned such as continuity, compactness, etc., are precisely why we need a topology to define them in the traditional approach. Since continuity, compactness, etc. can now be defined without open sets, your problem has been solved.

One of my favorite examples is the one-line proof of Cantor's intersection theorem via halos.

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  • $\begingroup$ I already noted that "So, we technically could say that a topology on X is a μ such that the open sets corresponding to μ 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 μ making no mention of open sets (and then open sets are later a defined concept, instead of given)" in the question. I'm looking for a definition directly in terms of monads. $\endgroup$ – PyRulez Feb 12 '18 at 23:15
  • $\begingroup$ I don't understand what you are looking for exactly. You want a definition "directly in terms of monads" of what exactly? The point is that for doing most applications of this circle of ideas, one doesn't need to speak about the topology explicitly at all, as illustrated by the example of Cantor's theorem. Similarly, when talking about continuous maps, homeomorphisms, etc. It can be done in terms of the halos. $\endgroup$ – Mikhail Katz Feb 13 '18 at 9:41
  • $\begingroup$ 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. $\endgroup$ – Mikhail Katz Feb 13 '18 at 11:04
  • $\begingroup$ 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). $\endgroup$ – Andreas Blass Mar 15 '18 at 0:56
  • $\begingroup$ 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. $\endgroup$ – Andreas Blass Mar 15 '18 at 0:59

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