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In Wikipedia, they say that a space is a set with some added structure. But what do they mean by "with some added structure"?

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There is no universal definition for "structure". –  Michael Greinecker Apr 23 '12 at 21:14
    
Look up topological space, or vector space, for two different kinds of structures. –  Álvaro Lozano-Robledo Apr 23 '12 at 21:15
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Was the link to the wikipedia entry on "space" not working? –  The Chaz 2.0 Apr 23 '12 at 21:16
    
In layman's terms, set = collection of elements. With added structure = operations on those elements + operations on subsets of those elements + properties which hold on those elements + properties which hold on subsets of those elements. So think of elements, subsets of elements, properties on those things, operations on those things. –  user2468 Apr 23 '12 at 22:28
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Wikipedia defines terms that have a clear mathematical meaning, and also terms that do not. This is one of the latter. It is best to pay no attention. –  André Nicolas Apr 23 '12 at 22:41

3 Answers 3

There are different kinds of spaces, therefore different kinds of structures. There is no such thing as "space" in general. You may consider a topological space, with a structure of open sets (fulfilling certain properties) - like a plane with open balls, its sums and so on. You may consider an algebraic object, like vector space, with algebraic structure, given by, for example, addition. You may actually have different structures on one space, and what you study are the interactions between them: the Euclidean plane has a both topological and algebraic structure, and these structures interact.

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Think of a game like pool. It consists of the following items - a pool table, 15 different colored balls, 1 white ball and a cue stick. Let us say that you don't know how to play pool. Now, I buy all the items mentioned above and give them to you, can you play the game of pool? You obviously can't because you don't know how these items are supposed to interact with each other. So, the next step would be that I explain to you how these items interact with each other and lay down some ground rules for a game of pool.

So, to learn the game of pool, I gave you two things - (collection of items) + (interaction explanation & rules).

A space is something like the game of pool. The set involved is akin to the collection of items above and the "structure" is akin to the explanation of interaction between the various elements of the set and some ground rules on those interactions.

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Great explanation –  user50229 Jul 16 '13 at 13:08
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@user50229 Thanks :-) –  TenaliRaman Jul 16 '13 at 14:25

There is a formal definition of structure, see here. In this definition, structure is clearly distinguished from property and from something else called stuff. This formalism may look highbrow, but on closer inspection it's not that complicated and makes a lot of sense.

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The problem is that there is more than one. Leo Corry wrote an interesting book on the history of the term in mathematics. –  Michael Greinecker Apr 23 '12 at 22:41
    
@MichaelGreinecker: I would argue that a variety of concurrent meaningful definitions is worth more than a vague metaphor. Could you point us to that book? –  Rasmus Apr 23 '12 at 22:53
    
I agree, but OP wrote as if there was some technical term that kept him from understanding what a space is. The book in question is Leo Corry, Modern Algebra and the Rise of Mathematical Structures. –  Michael Greinecker Apr 23 '12 at 22:55
    
@Rasmus: The trouble with this definition is that it is very, very relative, and at times conflicts with the intuitive notions. For example, the forgetful functor from the category of local rings and local ring homomorphisms to the category of rings is faithful, conservative, but not full – so by this definition it has forgotten structure. But I'm not wholly convinced that being a local ring involves extra structure! (The real reason for the definition, I think, is to make the category of affine schemes anti-equivalent to the category of commutative rings.) –  Zhen Lin Apr 23 '12 at 23:49
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@ZhenLin: Well being an object in the category of local rings with local ring homomorphisms involves extra structure! Being a local ring as an object of the category of rings doesn't. –  Rasmus Apr 23 '12 at 23:58

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