Surface

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).

Even simple surfaces can display surprisingly counterintuitive properties. For example, the surface of revolution of around the x-axis for (called Gabriel's horn) has finite volume but infinite surface area.