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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".2012-04-23
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    Look up topological *space*, or vector *space*, for two different kinds of structures.2012-04-23
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    Was the link to [the wikipedia entry on "space"](http://en.wikipedia.org/wiki/Mathematical_structure) not working?2012-04-23
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    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.2012-04-23
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    @J.D.: Properties are not the same as structures. For instance, an abelian group is *not* a group with additional structure. See also the link I posted in my answer.2012-04-23
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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.2012-04-23

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