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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.

  1. I am not sure how to think about this.
  2. 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.

sk8forether
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Make42
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  • Here's how I think about it. Suppose that $(X,d)$ is a metric space. So $X$ is a set, and $d$ is a function from $X \times X$ to $\mathbb R$ that satisfies certain properties. Often, the ordered pair $(X, d)$ is (perhaps confusingly) also called $X$. So the name "$X$" is being used for two different (but closely related) things, and the meaning of "$X$" depends on context. If someone says "$X$ is a metric space" then they are referring to the ordered pair. If someone says "Let $x \in X$", then here $X$ refers to the underlying set. This abuse of language is found to be convenient. – littleO Jun 04 '20 at 21:38
  • Another example: in my mind, $\mathbb R^n$ is defined to be the set of all ordered $n$-tuples of real numbers. But, the vector space $(\mathbb R^n, + , \cdot)$ is often also called $\mathbb R^n$. So the name $\mathbb R^n$ is being used to refer to two different things, and the meaning depends on context. If someone says that $\mathbb R^n$ is a vector space, then they're referring to the ordered triple. If someone says that $(1,2,3) \in \mathbb R^3$, then they're referring to the underlying set. – littleO Jun 04 '20 at 21:45
  • @littleO: I see your point - it is a little bit like an "implicit" in computer science / computer programming. I am also more inclined to say that $\mathbb R^n$ is not a vector space, but a set of tuples. So, you suggest to go with the abuse of terminology? Thing is, it gets somewhat weird, once you are starting to define something like a data space $D$ to be a tuple and then suddenly you are suddenly saying something like "the data object $o\in D$"... Or I would say the data space is actually as set, but then why do I call it a "space"? Why call the "sample space" a space...? – Make42 Jun 04 '20 at 21:50
  • This is a standard abuse of notation. It would be too annoying to write "Let $G = (S, *)$ be a group" and then always refer to group elements as elements of $S$ - you now have to keep two symbols in your head, while the abuse of notation lets you get away with one. – Jair Taylor Jun 04 '20 at 21:50
  • Really we are just overloading the meaning of the relation $\in$ so that $x \in G$ is short-hand for $x \in S$. – Jair Taylor Jun 04 '20 at 21:54
  • @JairTaylor: I could put this overloading into my "notations" section. It seems that the term "space" is also kind of heavily overloaded... https://en.wikipedia.org/wiki/Probability_space is even a triple (instead of a tuple) and the first element is a "sample space" which is a set, not a geometric space (as far as I can see for now)... – Make42 Jun 04 '20 at 22:01
  • @Make42 Yes, 'space' is not a well-defined term in math. – Jair Taylor Jun 04 '20 at 22:08
  • I should have written "instead of being a tuple with two elements" instead of "instead of a tuple". – Make42 Jun 04 '20 at 22:09
  • I am starting to think that I should abandon the idea of a space being a geometric space in the sense of being a tuple, where the first element is a set and the second some type of structure - even here "structure" does not seem to mean an algebraic structure. Instead, maybe I should think of a space as a "mathematical notion of space is a cavity where objects exist" https://math.stackexchange.com/a/2271640/340174 - so just "a thing where other things exist, like in a set, but a bit more, and not well defined"... :-D – Make42 Jun 04 '20 at 22:12
  • So, can I call anything like that a "xyz space" if I define it beforehand? – Make42 Jun 04 '20 at 22:13
  • I would just accept that if $(X, d)$ is a metric space, then the symbol $X$ is overloaded. So the underlying set and the ordered pair are both given the same name. That's not so bad, sometimes different objects are given the same name. I would embrace the abuse of notation. – littleO Jun 04 '20 at 22:14

1 Answers1

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"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."

R. Burton
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  • Very helpful! 1) I like you comparison to computer programming classes. 2) "At the same time, it isn't incorrect to refer to a general set as a "space," it just sounds weird." - well that has been done with "sample space", right? Or did I understand you wrongly? 3) Your last paragraph helped for my publication :-D - I am free to use "space" when naming my "collection type things". 4) You wrote that one can use the name of the underlying set to refer to a structure. But, more often I have a name for the structure and want to refer to the underlying set... how about that? – Make42 Jun 04 '20 at 23:00
  • @Make42 (in response to 2) Yes that's exactly what I meant. (in response to 4) there are a few options here, you could define an operation $set(S)$ which returns the carrier set of a structure $S$, or you could use a projection to get the first element of the tuple. I have seen $dom(A)$ used before to refer to the carrier set of a structure, and I think there are a few other similar notations. – R. Burton Jun 04 '20 at 23:06
  • It's also common-ish to use a calligraphic typeface for the structure and the same letter in the default typeface for the set - e.g. "a group $\mathfrak{G}=(G,\ast)$. – R. Burton Jun 04 '20 at 23:08