There is a formal definition of mathematical space?

Question

Yesterday I was searching for a formal definition of mathematical space, then I found it wikipedia article, where the definition rely in the concept of mathematical structure.

But I didnt found a formal definition of mathematical structure (it wikipedia article is very vague).

Then I dont know if exists a formal definition of mathematical space, or if this concept is just historical/utilitary.

If someone can clarify this question, through some reference or by any other way, I will be grateful.

Answer 1

No, there is no formal definition of the word "(mathematical) space", and this is not a term that is used with any precise formal meaning. The "definition" you found on Wikipedia is a definition in the sense of a dictionary entry, not a definition in a mathematical or logical sense.

(To be clear, there are many types of "spaces" that do have formal definitions, like vector spaces, topological spaces, etc. But there is no overarching precise mathematical concept of "space" that these are special cases of or anything like that. You can probably find someone somewhere who has written down a formal definition of a "mathematical space", perhaps along the lines of Robert Israel's suggestion of an object in a concrete category. But this is not the way people commonly use the term.)

Answer 2

Category theory is an attempt to formalize the notion of mathematical structure. So you might consider a "space" to be an object in a concrete category.

Answer 3

I would say that most things called a "space" in mathematics is either a vector space of some sort, a topological space of some sort, or both. My interpretation agrees with the hierarchy diagram on the Wikipedia page for "space".

A topological space is just a set (whose elements are called "points") and a consideration of some subsets of the set to be "open" (think unions of of open-disks in $\mathbb R^n$) where unions and finite intersections of open sets are open.

A vector space is just a set (whose elements are called "vectors") where you can add and subtract vectors and multiply them by numbers (elements from any field, technically, but $\mathbb R$ and $\mathbb C$ are common) and the operations act and interact in the way you would hope.

Every other space builds on one or both of these ideas. For example, a "normed vector space" adds in the concept of the "length" of a vector to vector spaces. A "metric space" adds in the concept of "distance between points" to a topological space.

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There are other instances of the word "space" that are not really referenced on the wikipedia page mentioned by the OP: "probability space" and relatedly, "sample space". Also, "measure spaces".