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I sometimes hear that "$A$ is a property of $V$, not a structure."

But I am not sure how to distinguish "property" from "structure" in general. Could you explain the difference with some easy examples?

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A property is something that is true of some object, it's intrinsic to the object. For example, being of finite dimension is a property of a vector space: either it's of finite dimension, or it's not. You can think of it as some kind of adjective, if you want: in "a blue pen", "blue" is a property of the pen.

A structure is something that you add on top of the object. To continue the same example, given a vector space $V$ of dimension $n$ (the dimension is a property of $V$), you can choose a basis $(e_1, \dots, e_n)$ of $V$, and consider the new object $(V, (e_1, \dots, e_n))$, a vector-space-with-a-basis. It's now a different kind of object.

If you have a structure $A$, not all objects can be upgraded to be an object-with-structure-$A$. For example, without the axiom of choice, some infinite dimensional vector spaces don't have a basis. So you have a property of objects, which is "being able to be given the structure $A$". For example, "having a basis" is a property of vector spaces, it's an intrinsic feature of the space. But the specific basis that you choose is not something intrinsic to the vector space, as there a vector space typically has many different bases, and each one is a different structure that you can add to you vector space.

Now, when you have a property $A$ that you can give to some type of objects, sometimes one can say "$A$ is a property, not a structure, of $V$". This means that $V$ can be given the structure $A$, and moreover it can be given that structure uniquely. So the structure that you can add to $V$ is actually intrinsic to $V$, there is only one possible given a $V$.

Here's an example: given a vector space, you can wonder whether or not you can make it into a topological vector space. It's not always possible in general, and if you can do it, there are several inequivalent ways of doing it. But when your space $V$ is of finite dimension, it's always possible to give a topology to $V$ that makes it into a topological vector space, and moreover that topology is unique (cf. this question). So what was a priori a structure on vector spaces becomes a property on finite dimensional vector spaces.

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  • $\begingroup$ Wonderful answer. $\endgroup$ – Arrow Nov 4 '17 at 14:50
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Let $V$ denote all nilpotent Lie algebras over a field $K$ of a given dimension $n$. This is sometimes called the "variety of nilpotent Lie algebra laws", although it need not be irreducible. Indeed, (ir)reducibility is a property of $V$, not a structure. On the other hand, $V$ has the structure of an affine algebraic set.

Another example is a given Lie group. It may have several properties, like being connected, simply connected, simple, semisimple, solvable, nilpotent, abelian, or something else, but these are no structures. But it carries a smooth manifold structure, and may carry several interesting geometric "structures".

So in general, a structure is really a well-defined additional structure, whereas "property" is quite general, and just means that any mathematical definition which can be applied to the object is either true or false (like a group is solvable or it is not solvable).

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I'm not sure what context you are interested in but maybe this example helps:

Take the set $\mathbb{Z}$. We can put a group structure on $\mathbb{Z}$ via the binary operation $\mathbb{Z} \times \mathbb{Z} \xrightarrow{+} \mathbb{Z}$ defined by $(n,m) \mapsto n+m$, so we could say $+$ gives a group structure on $\mathbb{Z}$. A property of $(\mathbb{Z},+)$ would be something like 'the sum of two even numbers is even'.

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