Difference between “space” and “mathematical structure”?

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

In my opinion, structure is the most general way of adding substance to a set. A space usually refers to something which has been topologized, is one way of adding structure.

Mathematical structure then really comes in two forms - algebraic and topological. Topological structures let you tell how your set is pieced together by special subsets, and so tells you something about the relationships between parts and the whole of the set. Algebraic structure tells you how the elements can interact with each other, and so in some sense tells you how parts relate to other parts.

In some sense then, these 'structures' are the scaffolding on which mathematics is built. It is the language we use to describe mathematical phenomena, and equips us with a way of reasoning about some abstracta, and often times just investigating the relatively simple aspects of our scaffolding can be a huge task. For this reason, it is quite literally the 'structure' of the subject.