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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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    $\begingroup$ 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. $\endgroup$
    – Stefan
    Aug 2, 2012 at 9:42
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    $\begingroup$ 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. $\endgroup$
    – Joachim
    Aug 2, 2012 at 9:50
  • $\begingroup$ 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. $\endgroup$
    – Joachim
    Aug 2, 2012 at 9:51

5 Answers 5

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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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  • $\begingroup$ 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. $\endgroup$ Aug 2, 2012 at 9:50
  • $\begingroup$ @Kevin Carlson However, many topology spaces have lost geometric property, for example, the discrete space. $\endgroup$
    – Paul
    Aug 2, 2012 at 9:52
  • $\begingroup$ 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. $\endgroup$ Aug 2, 2012 at 10:03
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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.

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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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  • $\begingroup$ 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. $\endgroup$
    – robjohn
    Oct 17, 2014 at 21:09
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I am a Wolfram language fan. I know that is not Mathematics but it offers some impressive structures. For example this relationship graph. The graph shows relationships between space classifications

enter image description here

This shows not that set is already a structure with objects. A set can be as complex as the Tarski-Grothendiek universes are. There are attributes given to space classes that are taxonomical like compact, complete, local, convex, and quasi-. The quasi- prefix shows up a great dilemma in classifying spaces. There is no need in properness, soberness or alike. The orientation is unsharp.

This graph is corrupt and damage. Each arrow is expected to show a label. By this fact another problem in presenting knowledge in the orientation of systematization there is not enough time to gather them all and there are so many properties available. The attempt not to work with names but for example with popular examples runs in even more trouble because the definitions are clear the popular examples are usually not. And a lot of well distinguished spaces are not popular and therefore not easy to be understood in the orientation of use.

The graph represents the Mathematical Notion of space. The physical notion of space is different to that.

The human nowadays ideas of space are highly diverse and include a lot of contradictions and incompatibilities. So arising from the presence of objects or ideas for mathematical methods a set and a space can be everything. Even the idea of empty space is included despite not the graph.

The causes are mathematically hard and often use logic even specialized logic in involved. The most abstract ideas are that stemming from structuralist like Bourbaki. These are under extended critics because structure is not always something regarded as positive or desirable. It may hinder solving problems or even stopping solutions. Space, sets, structure are coming together and in the moment with can distinguish these the problems are present. Empty, finite or in several kinds infinite space is causing trouble.

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