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For me its clear how to build up the object $S^1 \times S^1$ , who is our old friend torus, but the product of 'interior'of these objetcs doesn't look clear to me how to build up

I tried 'forget' one point of disc and then stretch the disc into a cilinder, where I glue to every point a disc and then glue the both 'mouths' of cilinder. The resulting object looks like a lot of torus hanging along another torus. But I know that must be wrong, since the product of two real manifolds with border is likely to not be a manifold. Is there a problem in the contact of these torus, the 'base' and the 'hangers', but I don't know if this is suficient.

Is There a kind of method to visualize such objects whom are made of manifolds with border ?

Thanks in advance

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The torus as a product of 2 1 dimensional manifolds embeds in 3 dimensional space, but as the disc is a 2 dimensional manifold, its product will not embed in 3 dimensional space but will necessitate higher dimensions.

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