Set vs abstract space: what's the difference? According to wikipedia, roughly:


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*A space is a set with some attached structure (measure, order, etc).

*An abstract space is a set with no structure attached


What is then the difference between a set and an abstract space? Is it the same as the difference between a pipe (abstract space) and a picture of a pipe (set)?
 A: I think you've misinterpreted that wikipedia article that you linked to in the comments. It does say that a "space" is a set with some added structure; but it does not say that an "abstract space" is a set with no structure attached. The first statement is a useful heuristic for thinking about commonalities among the various types of spaces that one encounters (vector spaces, metric spaces, topological spaces, Hilbert spaces, measure spaces, etc.). But the second statement is just false. An abstract space is, in particular, a space -- which means that it will always have some added structure in addition to being a set.
There is no rigorous general mathematical definition of either term, "space" or "abstract space." If you're having trouble wrapping your head around the concept of "abstract space," perhaps a useful way to think about it is this: Once you understand the definition of a vector space, then all the theorems about vector spaces tell you useful information about every concrete example you come across ($\mathbb R^n$, the space of all polynomials in one real variable, etc.). But if you want to prove something about all vector spaces, then your proof is going to have to apply to an arbitrary vector space, without knowing anything about how it is specifically constructed. Thus, you might say "Let $V$ be an abstract vector space," meaning that it is some set with added structure that we know satisfies the definition of a vector space, but we know absolutely nothing else about it. 
