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I am trying to understand the difference between a "space" and a "mathematical structure".

I have found the following definition for mathematical structure:

A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of associated mathematical objects is called the structure and the set is called the underlying set. http://www.abstractmath.org/MM/MMMathStructure.htm

Wikipedia says the following:

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance. http://en.wikipedia.org/wiki/Mathematical_structure

Regarding a space, Wikipedia says:

In mathematics, a space is a set with some added structure. http://en.wikipedia.org/wiki/Space_(mathematics)

I have also found some related questions, but I do not understand from them what the difference between a space and a mathematical structure is:

difference-between-space-and-algebraic-structure

what-does-a-space-mean

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Leaving aside that these terms are used varyingly and not precisely defined, your third quote pretty much states that a structure is part of a space, and that description is as good as any. –  Stefan Walter Aug 2 '12 at 9:42
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It might very well might be that there are multiple definitions, or none at all. Personally i think of a space as a set with a topology; i.e. a topological space. A "mathematical structure", well, isn't that just something you study in mathematics? ;) As for algebraic, note that groups, fields, rings, modules, vector spaces all are sets with some operations on them, so that could define an algebraic structure. Lastly, note that not everything in math has a set as underlying structure. –  Joachim Aug 2 '12 at 9:50
    
By the way, one nice and subtle example of the last thing i mentioned is that there is no forgetful functor from the category of (topological spaces and homotopy classes of maps) to (sets).. If you want i can elaborate on this. –  Joachim Aug 2 '12 at 9:51

3 Answers 3

up vote 5 down vote accepted

Neither of these words have a single mathematical definition. The English words can be used in essentially all the same situations, but you often think of a "space" as more geometric and a "structure" as more algebraic. The best approximation to a general "space" for many purposes is a topological space, but Grothendieck generalized further than that, to what are called topoi.

In model theory a "structure" is a set in which we can interpret some logical language, which is to say a set with some distinguished elements and some functions and relations on it. Some of the most common languages structures interpret are those of groups, rings, and fields, which have no relations, functions are addition and/or multiplication, and distinguished identity elements for those operation. We also have the language of partially ordered sets, which has the relation $\leq$ and neither functions nor constants.

So you could think of "structures" as places we do algebra, and "spaces" as places we do geometry. Then a lot of great mathematics has come from passing from structures to spaces and vice versa, as when we look at the fundamental group of a topological space or the spectrum of a ring. But in the end, the distinction is neither hard nor fast and only goes so far: many things are obviously both structures and spaces, some things are not obviously either, and some people might well disagree with everything I've said here.

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One example of a structure which I have never heard called a space is a graph: The point being that there are two sets of interest: The nodes and the edges. Depending on circumstance, one sometimes focuses on one or the other, and it is totally non-obvious which set should be considered the underlying set of a putative graph-as-space. –  Harald Hanche-Olsen Aug 2 '12 at 9:50
    
@Kevin Carlson However, many topology spaces have lost geometric property, for example, the discrete space. –  Paul Aug 2 '12 at 9:52
    
Paul: it's true that the geometry of the discrete space is uninteresting. But it does at least have a (trivial) geometry, in the Erlangen sense: its automorphisms are the trivial group, as the "geometry" of the indiscrete space are all its set-theoretic automorphisms. It's not significant geometry, but these are edge cases. –  Kevin Carlson Aug 2 '12 at 10:03

I'm not sure I must be right. Different people have different ideas. Here I just talk about my idea for your question. In my opinion, they are same: the set with some relation betwen the elements of the set. Calling it a space or calling it just a mathematical structure is just a kind of people's habit.

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A mathematical structure is a set or sets associated with some mathematical object (s) like a binary operation,collection of its subsets etc which satisfy some axioms.the mathematical object (s) is called structure and the set is called ground or underling set.example topological structure (X, tau) here tau is structure ans x is underlying set... similarly algebraic structure (x,*) now a space is a mathematical structure where the structure is of geometric type .for example (x,tau) here tau is a geometric type structurer so it is a space called topological space.

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This is a duplicate of your answer to Difference between "space" and "algebraic structure". Either this question is a duplicate, or your answer should be customized to answer this question. In either case, a duplicate answer is usually the wrong thing to do. –  robjohn Oct 17 at 21:09

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