Difference between "space" and "mathematical structure"? 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

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.
