Teil vs Unter for Mathematical Structures

I was always under the impression that Teil- is relating to an unstructured thing, and Unter- relates to a structure, i.e. usually a set of entities together with the relation(s) that define the particular structure.

That matches how the words are used generally in German (outside of mathematics). Unter- implies some sort of hiearchy, something you would represent in a vertical order, while Teil- simply refers to a share, without any particular shape or hierarchy. (What "hierarchy" means in those mathematical structures that use Unter- as prefix can, of course, be expressed formally.) In structures that have the Teil- prefix, any subcollection can be used. In structures with Unter- prefix, only those subcollections of the original collection can be used that maintain the structure of the parent collection.

That explains Teilintervall, Teilfolge, and Teilmenge vs. Untergruppe et. al.

I would explain your struggle with the usage in topology in the following way: When one uses Teilraumtopologie, one focuses on the fact that the subtopology is a subset of the parent topology. When you refer to the same thing as Unterraumtopologie, I would say you focus on the topology of the subspace, i.e., the subset together with the subtopology. To me, it seems this is about the order of linguistic derivation: Unterraum-topologie vs- Teil-(raum)topologie. The subtlety has a degree that it seems fair to me to call the difference irrelevant. I think it is notable here that a subset with the induced topology is always a topological space (again), while an analogical claim would not be true about algebraic structures like groups, rings, fields, and so on. That opens up this possibility of looking at the same thing from two slightly different angles. In a group, you can't just say: "Take the elements of the subset, together with the (reduced) group relation", because in the general case, this won't be a group anymore. Hence, considering a subgroup merely a subset of elements does not make too much sense. In a subgroup (subring, submodule, subfield, etc.) the whole point (the "thing one is interested in") is the fact that the relation of the parent works out on the sub-entity. That emphasis is not there when you talk about a subtopology.

I think, this reasoning can be expanded to other occurences of Teil- prefix where a structure is present. I would say that in those cases the emphasis is on the subset, not on the structure. I was surprised to read Teilvektorraum as a synonym in Wikipedia's definition of Untervektorraum - I have never heard of it when I studied mathematics (but I am not an expert, I stopped dealing with mathematics 15 years ago, and I was never working in acedemia there). But it could make sense if you think of the fact that the basis of any sub-vectorspace is a subset of a basis of the parent vector space. So, in that sense, if you characterize a vector space as a basis together with the field, it makes sense to think of a sub-vectorspace as a subset of a basis together with the same field.

With that, there remains the open question why the word Teilkörper exists. While Unterkörper does coexist with Teilkörper, a particular reason for the usage of Teilkörper might be that Unterkörper also means "lower body" -- and that sounds a little funny.