Understanding Surface Dimensions for Beginners

[Red_CCF]

How do we define the dimension of a surface? I know surfaces are 2-D but I don't really get where that comes from.

 

[Office_Shredder]

There are a couple of ways to define dimension. One common one (which explains why surfaces are 2 dimensional) is that around any point if you take a small enough open subset of your surface, there is a homeomorphism (a continuous bijection with a continuous inverse) from that piece of the surface to an open subset of the plane (or R³ or some other power of the reals depending on the dimension of your set). This is the idea of a manifold:

http://en.wikipedia.org/wiki/Manifold

For a surface, you probably have a parametrization which basically describes how to form these homeomorphisms either immediately or with only a little bit of work

 

[HallsofIvy]

Perhaps more importantly, how do you define "surface"?

 

[Red_CCF]

Perhaps more importantly, how do you define "surface"?

I don't really have a formal definition, but I would say something like a plane or a hollow sphere on R3 (ex. x^2+y^2+z^2 = k) is a surface.