Topology as an Algebraic Structure 3

In this post, we continue the exploration from the previous in the series.

As discussed in the exploration on formalization of equivalency of axiomatizations, we’d want to basically work with structure-preserving maps (homomorphisms) here. In that post, I raised a question about motivation for representation theory as expressed in the Wikipedia article on the subject, but even if we concluded that a homomorphism from a set of matrices to an “unfamiliar” set is what is needed to yield statement implication on the unfamiliar set, not the other way around, it still may be possible to glean some information from a reverse construct (unfamiliar set to set of matrices) as the Wikipedia article implies. Plus, the term “representation” suggests that the unfamiliar set is being “represented” as a set of matrices, implying a map from the unfamiliar set to the set of matrices (that presumably respects some structure), not the other way around. I’ll discuss this further in another post on different motivations for representation theory.

But for now, we will take the approach of using homomorphisms from the familiar set to the unfamiliar set — so the reverse of a representation.

So let $T$ be a topological space; we want to study $T$ . Thus, $T$ is the unfamiliar set. Then, the familiar set is some algebra $A$ (a set $A$ with an algebraic structure on it — a collection of operations satisfying some axioms. That concept can be formulated in say first-order model theory, and we can then restrict to first-order model theory without relations, aka universal algebra. The collection of operations plus axioms for those operations would be called a language, $L$ .) (Note: I had originally called this a signature, but after learning more model theory I’ve been using the term language instead, so we will use “language” here.) So given $T$ , we can define a universal algebraic correspondent of it to be a set $A$ , a language $L$ , and a homomorphism from $A$ to $T$ . What structure should the homomorphism respect?

If we can vary the language $L$ and have just a homomorphism, then that would be the most general case, but on second thought that might seem too general for us to study now. We could restrict to either not letting $L$ vary or replacing “homomorphism” by “isomorphism”; for now, since we already did some work with bijections that suggested the isomorphism route, let’s study that first. (Although the general formulation will still be homomorphism from potentially varying $L$ .)

Thus, we will for now restrict attention to: given a topological space $T$ , we consider correspondents that consist of a set $A$ , a language $L$ , and an isomorphism from $A$ to $T$ . (In keeping with the terminology for representations, we can call a correspondent with an isomorphism a fully faithful correspondent. Thus, we will study fully faithful correspondents of topological spaces.) We showed earlier that, assuming $L$ consists of finite arities, we cannot have a constant $L$ for different $T$ ; this was purely off of cardinalities and no other structure that was respected. But what structure should such an isomorphism respect? How should we define this?

Presumably, we want to see common operations or possibly even relations in topology, and then require that those are preserved. In fact, I’m confident we can generalize this discussion: if we generalize the language $L$ to be any first-order language, then I’m confident we can yield (off of the same cardinalities argument as before) that $L$ must vary if we are considering fully faithful correspondents. With that the case, we can generalize beyond universal algebra and try to use first-order model theory to study topology: we can define a first-order correspondent in general to be, for a given topological space $T$ , a tuple of a set $M$ , a language $L$ , and a homomorphism from $M$ to $T$ . Really, this all depends on what structure we want to respect in $T$ — identifying operations or possibly relations and requiring that those are respected. If we don’t identify any relations, then the generalization to first-order structures is meaningless, otherwise it can yield fruit. So let’s turn our attention to identifying common operations and relations that are done in topology — let’s start with operations first. (So we’re looking at algebraic correspondents. Written out fully, we’re looking at fully faithful algebraic correspondents.)

Topology is defined by unions and intersections of open sets; are these operations on open sets? No, because the arities change, especially for arbitrary unions. But actually, we already have a framework for that — arities can change because the language $L$ depends on the space $T$ . So we could presumably define our required language $L$ to contain arities corresponding to the possible unions of open sets that can be taken, and then we require the union to be preserved. In other words, $L$ could contain an operation of each arity up until and including the cardinality of $\tau$ (the topology on $T$ ); then, for any subset $\tau^{'}$ of $\tau$ , we could require that for the operation $o$ in $L$ of cardinality $\left| \tau^{'} \right|$ , our isomorphism $f$ satisfies $f\left( o\left( a_{1},\ldots \right) \right) = \cup \left( f\left( a_{1} \right),\ldots \right)$ , so that essentially $f$ “maps” $o$ to union “in a preservation sense.”

If we look at measure theory and measurable spaces, we could possibly do this while allowing $L$ to stay constant, since measurable spaces only require closure under countable unions. Thus, $L$ could consist of arities $1,2,\ldots$ up until $\aleph_{0}$ (basically, the set of arities of $L$ is $\mathbb{Z}_{+} \cup \left\{ \aleph_{0} \right\}$ ), and each of these could be required to be preserved in accordance to a union in a “natural” way similar to the previous formulation. It might be a bit easier to study non-varying $L$ first, so we could possibly take measure theory as a guinea pig before looking at topology.

It remains to study this further.

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