What is a surface in Maths and/or Physics?

I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

Can this be generalized mathematically as we introduce more (higher) dimensions?

Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

*ie as a limit