Equipping model-theoretic structures with topologies -- what is it called? how do you handle predicates?

Question

Equipping $L$-structures with topologies.

It seems like we can build interesting structures by equipping the domain of structures with topologies and requiring the interpretation of function symbols to be continuous with respect to the topology.

This seems like an obvious thing to do in order to limit how "wild" a structure can get.

• Does this construction have a name?
• What constraints are typically placed on the interpretation of predicate symbols in this setting and why?

My question is similar to this question but is explicitly about the interpretation of predicate symbols as well. Also, the linked question seems to be about examining particular theories in detail.

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Consider $\mathbb{R}$ equipped with its standard topology $\tau$. Consider the language $L$ with symbols $\{+, -, *, 1, 0\}$ and the theory of commutative rings with identity.

We can build a trivial $L$-structure by taking the domain in our structure to be $\mathbb{R}$ and giving each of our function symbols their usual interpretation.

The unary function $-$ is continuous.

For $+$ and $*$, the inverse image of a closed set in $\mathbb{R}$ is closed in $\mathbb{R}\times \mathbb{R}$.

So, our trivial model (which is really just $\mathbb{R}$ thought of as a ring) respects the underlying topology associated with $\mathbb{R}$, which is nice.

The continuity restriction stops us from producing a new equivalent-but-technically-different model by relabeling every element of our domain.

I'm curious what happens if we have predicates as well.

We can do something very ad hoc and insist that predicates be continuous maps into the Sierpiński space $\{\varnothing, \{0\}, \{0, 1\}\}$ and insist that the inverse image of $\{1\}$ is closed in our topology.

If we do this, I think $\le$ would be allowable as a predicate but $<$ would not be, since I'm pretty sure that $(<)^{-1}(\mathrm{true})$ is open in $\mathbb{R} \times \mathbb{R}$.

Anyway, that feels extremely ad hoc, so I'm wondering what the real way of resolving this problem is.

Answer 1

One paper you might be interested in is Pillay's First Order Topological Structures and Theories. This paper surveys a few different notions of topological model:

(i) $M$ might be a model whose underlying set happens to have a topology, but the model theoretic operations/relations don't need to have anything to do with the topology. Imo this is the least interesting option explored, and I think the author agrees.

(ii) View $M$ as a model of the signature you expect, plus a new (monadic) second order predicate telling you which subsets are open. This is definitely interesting, but isn't really in the spirit of your question.

(iii) Let $M$ be a model, and for some formula (with parameters) $\phi(x, \overline{a})$ consider the topology generated by $\{ \phi(x,\overline{a})^M \mid \overline{a} \in M^n \}$. This, I think, is the most in line with your question. In fact, the author gives a few examples of ways to get traditional topological structures with this machinery.

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If that isn't entirely satisfactory (it wasn't for me), you might want to look into Categorical Logic. In particular, what you're describing is called "doing model theory internal to $\mathsf{Top}$", and that kind of query is likely to bring up examples (e.g. "Groups internal to $\mathsf{Top}$").

The idea with the categorical setup is to think about doing model theory inside of any category. Then you want your function symbols to be given by arrows in your category (so inside of $\mathsf{Top}$ all our operations must be continuous) and you want your relation symbols to correspond to subobjects in your category (so in $\mathsf{Top}$, a relation might be any continuous injection into your model. It need not have the subspace topology).

This is nice because it unifies a lot of the "variants" of models that we see in the wild. A topological group is a group internal to $\mathsf{Top}$, a Lie group is a group internal to $\mathsf{Diff}$ (the category of smooth manifolds), an algebraic group is a group internal to a category of varieties, etc.

Of course, we need our category to have quite a bit of structure if we want to be able to interpret lots of formulas. As a simple example, if $R$ and $S$ are two predicates in our language, when can we interpret $R \land S$? Well that formula should correspond to the meet of $R$ and $S$ in the poset of subobjects of $M$, and so we're forced into requiring some structure on the category $\mathsf{Sub}(M)$.

Weaker logics can be interpreted in more categories, but stronger logics can say more things, so there's a fun balancing act we get to play. Unfortunately, $\mathsf{Top}$ is a kind of crummy category, but if we look at, say, the compactly generated hausdorff spaces we recover most of the spaces we actually care about, and we get a regular category, which has enough structure to interpret regular logic. There's a whole slew of variations on $\mathsf{Top}$ all of which have nicer categorical properties. You can find a list here, and they all have pros and cons.

The most structure you could ever want a category to have is captured by the idea of a topos, which lets you interpret anything your heart desires (including higher order logic!). Again, people have tried to find toposes which are related to $\mathsf{Top}$, and you can find a survey here.

If you want to read more about categorical logic, you might try Makkai and Reyes' First Order Categorical Logic. For a more modern, and slightly more leisurely, reference you might try Steve Awodey's lecture notes here.

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I hope this helps ^_^