I am kind of confused on the terminology of "space". From https://en.wikipedia.org/wiki/Space_(mathematics) I am getting that
In mathematics, a space is a set (sometimes called a universe) with some added structure.
And from topological and metric spaces, I know that we a space is a a tuple of a set and a structure, e.g. $(X, \tau)$, $(X, d)$, where $\tau$ and $d$ are a topology and a metric respectively.
On the other hand, in machine learning the term "features space" is used a lot for sets like $\mathbb R^n$, e.g. here and although it often refers to the set underlaying set itself, not the tuple of set plus the added structure. In fact, I have done the same in a previous publication (which the peer-reviewers accepted), but I would like to be both correct and precise in what I research, write, and submit. Is it just that machine learning researchers are imprecise in their terminology? Is it just an abuse of terminology?
I think, what some people are doing, might be they consider a space to be a set, which is somewhat structured, instead considering the set with the structure (so, the tuple), to be the space.
- I am not sure how to think about this.
- How do I deal with this in my writing? Especially since, what happens a lot is that I need subsets and element out of the underlying sets of all kinds of spaces.
Of course it would greatly help if there was a general name for a set that is the underlying set of a space, for which I asked, but it seems that there is no dedicated name for such a set.
Let's say there was a features space $(X, \cdot)$, (where I am not even sure, what structure we would add). Then it would be great to have a name for the set, let's call it an asdf. So we could say the feature asdf.
Afterthoughts:
What makes it worse for me is that it seems (https://math.stackexchange.com/a/174297/340174 and https://math.stackexchange.com/a/177943/340174) I am not even using the word "structure", right, since it is about operations, so we are talking about an "algebraic structure", while "geometric space" is... something else...? So apparently a "vector space" is actually not a "geometric space", but an "algebraic structure". I can understand that point, but the language gets even more confusing.
"Space," "set," "structure," and "model" are each used interchangeably at some point across the various fields of mathematics. The imprecise terminology is not unique to any one field, and much of the confusion stems from historical usage (some terminology predates formalization).
In general, the following conventions may be observed in various fields:
A "set" is a type of container. In common usage, a set may also have additional properties (such as a function or relation being defined on it) while still maintaining its "setness" (i.e. $\Bbb{R}$ is regarded as a "set" rather than an "algebra" or "theory").
A "structure" is an $n$-tuple consisting of a set, one or more functions, and one or more relations defined on that set. In practice, "structures" behave similarly to classes in computer programming.
A "model" is the model-theoretic conception of a "structure".
A "space" is a set, and possibly, but not necessarily, a structure. Honestly, I don't think that the term "space" is meant to have a specific meaning, since the things called "spaces" do not necessarily have anything in common. I would consider that "space" is most commonly used in reference to something implicitly considered to be a topological space (e.g. $\Bbb{R}^n$ is usually treated as $\Bbb{R}^n$ + the Euclidean topology, even when this is not stated). At the same time, it isn't incorrect to refer to a general set as a "space," it just sounds weird. There are also cases of "space" that are unrelated to topology (probability space comes to mind); although it isn't too difficult to relate almost everything to topology in some way if you try hard enough.
From a linguistic standpoint, the terms "space," "set," and "structure," as they are generally used, are related by:
space < structure < set (< = is a hyponym of).
Depending on who you ask a "set" may also be a structure - albeit a trivial one - in which case "set," and "structure" can be used interchangeably, leaving "space" to refer to non- trivial structures.
As stated in comments, it is also common to abbreviate a structure using the name of the carrier set. In some contexts (e.g. applied mathematics, analysis, number theory), particular "sets" (e.g. "the reals") are defined as a particular. For example, the statement "every real number greater than $0$ is the square of another real number greater than $0$" only makes sense if an operation - "square" - is defined. Since "operations" are only meaningful in the context of "structures", this means that "the real numbers," as we know them, behave more like a "structure" than a "set." Despite this, the thing denoted by "$\Bbb{R}$" is generally regarded as a "set" rather than a "structure."
The Formal Distinction
There are formal definitions for the term "set" and "structure" in mathematical logic and foundations. The exact definition depends on your choice of foundations.
In general, a "set" is any term of a "set theory" (e.g. ZFC, NBG, etc.) which is not a proper class (when proper classes are present). It is possible to encode "sets" in other systems as well - for example "sets" can be assigned to a type in type theory or a category in category theory (the category of sets).
A "structure" is set $S$, along with a set of operations $S^n\to S$ and relations $\subseteq S^n$ defined on $S$. By definition, a structure is also a set if you are using a pure set theory as your foundations.
In model theory, the terms "model" and "structure" can be used interchangeably. For a formal overview of "structures" as it applies to model theory, see Weiss - Fundamentals of Model Theory. The only major difference between "structures" in algebra (and, to an extent, category theory) and "structures" in model theory is that algebra typically regards the structure as an entity unto itself, while model theory treats structures as "models" of formal theories.
The term "space" does not have a formal definition as far as I am aware and do not know of any "space theory". On the basis of usage, I would say that pretty much anything that is "sufficiently set like" (i.e. not a large category or proper class) can reasonably be called a "space."