The word "surface" is an important term in mathematics and is used in many ways. The most common and straightforward use of the word is to denote a two-dimensional
submanifold of three-dimensional Euclidean
space. Surfaces can range from the very complicated (e.g., fractals
such as the Mandelbrot set) to the very simple
(such as the plane). More generally, the word "surface"
can be used to denote an -dimensional
submanifold of an -dimensional manifold, or in general,
any codimension-1 subobject in an object (like a
Banach space or an infinite-dimensional manifold).
-dimensional
submanifold of an 
-dimensional manifold, or in general,
any codimension-1 subobject in an object (like a
Banach space or an infinite-dimensional manifold).
around the x-axis for
(called Gabriel's
horn) has finite volume
but infinite surface area.
