It seems like there's a space for everything, functions, matrices, complex numbers, so on. Because I see this term "space" used so often, I have to ask: when is a set of mathematical objects a "space"?
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I don't think "space" by itself is defined. Much like, "number" by itself isn't defined. See https://en.wikipedia.org/wiki/Space_(mathematics) – ShawSa Aug 31 '22 at 00:19
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While it isn't clear-cut, in practice, I feel that we sometimes use the term when there's some sort of geometry or topology involved. In particular, you may consider the same object as a space in some contexts, and not as a space at other times — for example, when I work with the integers as a discrete, perhaps combinatorial kind of object, I don't really think of it as a space. But if I'm working with the integers as a dynamical system, I'm more inclined to think of it as a space. I get an intuition of points moving around, which reminds me of real space. – ho boon suan Aug 31 '22 at 00:31
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Most of the time, when mathematicians use the term “space”, they are either talking about a vector space, a topological space, or some combination of the two (like a normed vector space, for instance). But the word “space” itself doesn’t have a meaning. – Mark Saving Aug 31 '22 at 02:05
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In addition to those already mentioned, nother usage is probability space and the related “sample space”” That seems to cover the common uses. – BrianO Aug 31 '22 at 05:35