A definition of topology using monads (a.k.a. halos)

Question

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).

Answer 1

Two researchers in the 1980s have independently discovered the necessary axioms for defining a topology out of halo axioms. The relevant papers are:

Vakil, N. Monadic binary relations and the monad systems at near-standard points. The journal of Symbolic Logic, 52(3):689-697, Sep 1987. (link)

and

Tewfik, S. General topology. In Diener, F. and Diener, M., editors. Nonstandard Analysis in Practice, pp. 109-144. Springer-Verlag Berlin Heidelberg, Berlin, 1995. (link).

I accessed these resources via my university so you may not be able to see the material directly in the links above, unfortunately.

In these papers it is proved that in order for a standard topology to be uniquely determined, it is necessary and sufficient that the infinitesimal closeness relation of a space be monadic, locally reflexive, and locally transitive. I will elaborate on the meaning of these axioms a bit more.

Monadicity

This axiom requires that there must be a filter in $X \times X$ whose intersection monad is the infinitesimal closeness relation in *$X$. In other words, if every point $x$ in *$X$ has its halo $\mu(x) \subset$ *$X$, then the relation

$$W = \left \{ (x, y) \in \text{*} \left( X \times X \right) \mid y \in \mu(x) \right \} $$

should be able to be written as

$$W=\bigcap_{F \in \mathcal{F}}{\text{*}F}$$

for some filter $\mathcal{F}$ in $X \times X$. (In order to guarantee that all filters have a nonempty intersection monad, a model with sufficient saturation properties must be assumed.)

Tewfik chooses a slightly different approach and requires that the relation be expressed as a halic formula. Since I don't know much about IST I don't really understand his definition, but considering what he says in the book I believe it to be more or less equivalent to the one presented above.

Note that in order for the above axiom to make sense, the halo system must be defined for all points in *$X$, not only the standard ones in $X$.

Intuitively, this axiom says that if two points are infinitesimally close to each other then we should be able to approximate them with a collection of ever-decreasing sets, kind of like when we approximated with the collection $\left \{ x \in \text{*} \mathbb{R} \mid \lvert x-a \rvert < \frac{1}{n}\right \} \left(n \in \mathbb{N} \right)$ points infinitesimally close to $a \in \mathbb{R}$.

Local reflexivity

This axiom states that $x \in \mu(x)$ for all $x \in X$. Tewfik noted that this automatically guarantees reflexivity ($x \in \mu(x)$ for all $x \in \text{*}X$), so it is sufficient only to state the local proposition as an axiom. The intuition is obvious; we should be able to say of any point that it is infinitesimally close to itself.

Local transitivity

This axiom requires that for all $x \in X$, $y \in \mu(x)$ and $z \in \mu(y)$ should imply $z \in \mu(x)$. From this axiom we can deduce the fourth neighborhood axiom of topology (namely, that any neighborhood contains some smaller open neighborhood). This allows us to prove that if a halo system $\mu$ defines a topology $T$, then we can work backwards from $T$ to retrieve precisely the halo system with which we started: $$\mu(x)=\bigcap{\left \{\text{*}O: O \in T, x \in \text{*}O \right \}} \, (x \in X) \text{.}$$

Therefore we can be assured that every halo system satisfying the above axioms uniquely define a topology in the classical sense, so this nonstandard characterization of a topology really coincides with the classical one.

The intuition behind 'transitivity' is fairly acceptable, I think. From the notion of 'closeness' we get the impression that if some point $A$ is close to another point $B$ that is again close to $C$, then probably $A$ should be close to $C$ as well. For the 'local' part I haven't been able to find a persuasive justification, however.

This axiomatization is elegant, intuitive, and can be readily generalized to other settings (for example, it is well known that we can characterize uniform spaces by requiring its infinitesimal closeness relation to be a monadic equivalence relation). I think it deserves to be known better. I wonder why the relevant papers seem to have been largely forgotten or dismissed.

Edit:

Originally, there was a paragraph in the section 'monadicity' that said,

Alternatively, one could define halos $\mu(x)$ for only $x \in X$, state as an axiom that for each $x \in X$ there must be a (neighborhood) filter $\mathcal{N}(x)$ whose intersection monad is precisely $$\mu(x)=\bigcap_{N \in \mathcal{N}(x)}{\text{*}N}\text{,}$$ and then extrapolate this to define halos of nonstandard points $x \in \text{*}X$ as $$\mu(x)=\bigcap_{\text{*}N \in \text{*}\mathcal{N}(x)}{\text{*}N}\text{.}$$ (This is not Vakil and Tewfik's idea. I worked this formulation out just because I'm not used to dealing with binary relations, and also because I found it quite confusing that the definition of halos should include all nonstandard points.)

This formulation in general is not equivalent to Vakil and Tewfik's approach. According to their approach, multiple halo systems can define the same topology if they agree with one another on every standard point. My way of 'extrapolating' therefore amounts to arbitrarily choosing a single halo system for each topology. In fact, it is equivalent to requiring that $$\mu(x)=\bigcap{\left \{\text{*}O: O \in T, x \in \text{*}O \right \}} \text{,}$$ even for nonstandard points $x \in \text{*} X$. A fairly natural choice, but still arbitrary. If one wishes to replicate Vakil and Tewfik's condition, then one can impose the existence of a 'set of indices' $I$, and require that for each $x \in X$ its neighborhood filter admits a basis $N(i, x) \, (i \in I)$. Now if we extrapolate this to $$\mu(x)=\bigcap_{i \in I}{\text{*}N(i, x)} \, (x \in \text{*}X) \text{,}$$ one can easily show that $$W = \left \{ (x, y) \in \text{*} \left( X \times X \right) \mid y \in \mu(x) \right \} $$ is the intersection monad of the filter generated by the basis $\left\{(x, y) \mid y \in N(i, x) \right\} \, (i \in I)$. The result of the extrapolation will depend on the choice of basis, however.

Answer 2

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.