I was asked the question: "What is a space?". Wikipedia says it is a set with added structure, but then why don't we call a group a space, or a ring? The Princeton companion doesn't even have an entry for 'Space'. Where does the word 'space' come from? Who used it first? Is it perhaps a too common word in the English language? - In summary. What is the best answer to "What is a space?"
When dealing with these kind of questions, I like to consult the site of Earliest Known Uses of some of the Words of Mathematics. For the entry under "Space", it says the following:
SPACE. The word came into English—from Old French from Latin—around 1300. The OED [Oxford English Dictionary] entry distinguishes many meanings. In one sense (under heading 6b) it has room as a synonym. This word derives from the Old English and is related to the modern German Raum. Under heading 17 the OED defines “a space” as “an instance of any of various mathematical concepts, usually regarded as a set of points having some specified structure.” Among the quotations is a nice one from 1932: “The word ‘space’ has gradually acquired a mathematical significance so broad that it is virtually equivalent to the word ‘class’, as used in logic.” (M. H. Stone Linear Transformations in Hilbert Space p. 1.) The space age was well under way by 1914 when Hausdorff’s Grundzüge der Mengenlehre (Fundamentals of Set Theory) gave axioms for a METRIC SPACE (metrischer Raum) and for a TOPOLOGICAL SPACE (topologischer Raum).
See the entries BANACH SPACE, HAUSDORFF SPACE, HILBERT SPACE, METRIC SPACE, POINT, TOPOLOGICAL SPACE and VECTOR SPACE.
One thing I like about this entry is how it says that the word "space" in English is related to the German word "Raum", which means "room" in both the literal sense, but also the abstract sense of "a place in which something can reside", like "Zeitraum" or the more murky "Lebensraum".
With this interpretation, it makes sense to speak of spaces as a general term for "rooms of points" or perhaps "room for points", and when you add additional structure to your space, you simply specify the structure in the name of the new object, as in "metrischer Raum" and "topologischer Raum".
Why then isn't a group called a space? I think the reasons are two-fold:
First, it appears that the term "group" originated in France, as mentioned here, and the custom of naming everything "[some adjective] space" might not have been adopted by France at this time. Second, and I think this is the primary reason, groups didn't originate as sets of geometric objects, or sets of points, but rather as sets of symmetries. It seems unnatural to think of symmetries as being points residing in some space somewhere, which is why groups deserved to be their own entity.
Well there are several different things that you could describe as "a space". Most commonly (I find) a "space" refers to a vector space; probably named because the vectors space $\mathbb R^3$ describes three dimensional space that we inhabit (superficially of course, I can't comment on the actual dimensionality of the space we inhabit). A second common use for "space" is in reference to a topological space, where a set is given a notion of "closeness" in some sense.
N.B. There are probably numerous other uses, so one should always endeavour to be specific as to what kind of space you are dealing with.
A book was just released: Mathematical Concepts by Jurgen Jost with an entire chapter devoted to the question "What is a Space?".