Classification of topological points
I've recently dreamed up the following idea to classify topological points, and I'm wondering if this is or relates to a known concept. Here “topological point” refers to a point together with its neighbourhood structure (the definitions below should make clear what I mean).
Given a topological space $(X,\mathcal T)$, one can define a preorder $\precsim$ on its points as follows:
Given two points $p,q\in X$, we have $p\precsim q$ if there exists a neighbourhood $N$ of $p$ and a subset $S$ containing $q$ such that there exists a homeomorphism from $N$ to $S$ with their respective subspace topologies which maps $p$ to $q$.
Using this preorder, one can then define the obvious equivalence relation $p\sim q\iff p\precsim q\land q\precsim p$, resulting in a partial order on the equivalence classes.
Now looking at the definition of the preorder above, one notices that there is no inherent reason to restrict it to points of the same space; it works just as well if $p$ and $q$ come from different spaces (except that you run into issues of proper classes, but those can be handled in the usual ways). That is, it actually gives a classification of arbitrary topological points.
Some interesting facts (here the “type” of a point refers to the set representing its equivalence class):

If $p\in\mathbb R^m$ and $q\in\mathbb R^n$, then $p\precsim q$ iff $m\le n$. Obviously the same holds more generally for topological manifolds of dimension $m$ and $n$.

In a manifold with boundary, interior points and boundary points each form an equivalence class. The boundary points are of lesser type than the interior points, but still of greater type than the points of a manifold of lesser dimension (indeed, this is the case I originally considered when coming up with this classification).

The isolated point is the absolute minimal type of topological point. There does not exist a maximal type (indeed, the types themselves form a proper class).
My knowledge of topology beyond the basics comes mostly from the internet, so it is quite possible that I just rediscovered a wellknown concept, or some variant of a wellknown concept. If so, what is it called, and if it differs from my idea, how does it differ?
1 answer
This may not completely answer your question, but it sounds like what you're talking about is related to the concept of topological embedding. A function $f: X\to Y$ that is a homeomorphism from $X$ to $f(X)$ is called an embedding. For a given $X$ and $Y$, the existence of an embedding of $X$ in $Y$ is a topological invariant. This is a relationship between sets though. Your relationship could be stated in terms of embedding as follows:
Given two points $p\in X$ and $q\in Y$, we have $p\precsim q$ if there is an embedding $f$ from a neighborhood of $p$, into $Y$, with $f(p)=q$.
1 comment
Thank you. I did know the term embedding for Riemannian manifolds, but wasn't aware of the purely topological usage. Stated that way, the relation looks natural enough that someone ought to have thought of it before.
2 comments
This sounds almost like the small inductive dimension. — msh210 8 months ago
@msh: I've now looked up the small inductive dimension, and I don't think it fits. In particular, the point types in general are only partially ordered, while the small inductive dimensions are by construction totally ordered. Also, if you equip a vonNeumann ordinal $\alpha$ with the topology $\alpha+1$, then every point (i.e. ordinal) in $\alpha$ has a different type (ordered the same as the ordinals), but IIUC the inductive dimension is $0$ (because the closure of any open set is $\alpha$). — celtschk 8 months ago