## Is there a more user friendly approach to subsets in manifolds ?

In algebraic structures the substructures have a simple and clear definition. For instance, subgroup, subring, subalgebra, subfield, vector subspace, and so on. Manifolds are supposed to be but curved versions of such, so to say, straight or linear algebraic structures. However, when it comes to submanifolds and other more or less nice subsets, we have quite some complications. For instance, as subset can be an immersion, without being a submanifold. Or it can be a submanifold, without being an imbedding. Could one perhaps given another definition of manifold in which such complications do not arise, and we are back to the simplicity and clarity of algebra ?

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Are manifolds really "curved versions" of things like subgroups etc.? Perhaps the complexity of definitions of immersions, embeddings, submanifolds etc., reflect the greater complexity of geometry over algebra. – Robin Chapman Jul 12 2010 at 15:16
Even if you defined another class of objects for which those complications did not arise, in all likelyhood we would end up studying them anyways: it is not that we study immersions in all their complexity because we messed up when we defined manifolds and are forced to live with the consequences, but because we are interested in immersions and their complexity! You cannot define away our interest in them. – Mariano Suárez-Alvarez Jul 12 2010 at 15:20
It is actually a bad idea to view an immersion as a "subset", because the topology of the subset can create additional difficulties. It is better to study it as a map from one manifold to another. I believe, perhaps mistakenly, that in almost every subject it is more user-friendly to study maps between objects in a category than subobjects. This is definitely the case for manifolds and differential geometry. – Deane Yang Jul 12 2010 at 15:40
I mostly agree with Deane; however, there are definitely cases in topology where it would be a bit perverse not to think about submanifolds as subobjects (for instance, simple closed curves on surfaces or incompressible surfaces in 3-manifolds). – Andy Putman Jul 12 2010 at 15:53
And I agree with Andy! I tend to think about things from an analytic point of view, where functions and maps are usually much easier to work with than subobjects. But a geometric topologist often (usually? always?) works with subobjects instead of maps. – Deane Yang Jul 12 2010 at 17:01
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