Surface is a three-dimensional geometrical object that is defined by a set of points in a three-dimensional space. It is a topological space that is locally homeomorphic to a plane, but may not globally be a plane or even topologically equivalent to a plane.

Curves can be thought of as being parameterized by a single parameter, such as a line in two-dimensional space being parameterized by its length. Surfaces, on the other hand, are parameterized by two parameters, such as a plane in three-dimensional space being parameterized by its two coordinates.

One way to visualize a surface is to think of it as a sheet of rubber stretched over a three-dimensional object. The rubber sheet can be manipulated and deformed, but it always remains a single, continuous surface.

Surfaces can be described mathematically using equations and functions. For example, the equation of a sphere is given by (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a,b,c) is the center of the sphere and r is the radius. This equation describes a surface that is defined by all the points that are a fixed distance r from the center of the sphere.

There are many different types of surfaces, including planes, cylinders, cones, and spheres, to name a few. Each type of surface has its own unique properties and characteristics, and can be used to model a wide variety of real-world objects and phenomena.

In addition to being studied in mathematics, surfaces play a important role in many fields, such as engineering, computer graphics, and physics. Understanding the properties and behavior of surfaces is essential for many applications, including the design of aircraft and automobiles, the creation of computer-generated imagery, and the simulation of physical phenomena.