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I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different with respect to some property and I just wanted to know more about it.

EDIT: I try to add clearer informations about what I want to talk about. In practice, when we talk about some structured set, we can view the structure in more different ways (as lots of you observed). For example, when someone speaks about $\mathbb{R}$, one could see it as an ordered field with particular lub property, others may view it with more structures added (for example as a metric space or a vector space and so on). Analogously (and surprisingly!), even if we say that $G$ is a group and $G^\ast$ is a permutation group, we are talking about different mathematical object, even if they are isomorphic as groups! In fact there are groups that are isomorphic (wrt group isomorphisms) but have different properties, for example, when seen as permutation groups.

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    What property would they be different with respect to?2012-06-02
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    @Potato: a property which is not captured by the notion of isomorphism you're considering!2012-06-02
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    One example is topological spaces and metric spaces. There are metric spaces that are homeomorphic (topologically isomorphic, ie. same open sets) but that are not isomorphic as metric spaces (for example, one is complete and the other is not).2012-06-03

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