surface
Sign in to savetwo-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space
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A surface is a two-dimensional shape or space, like the face of a sphere or a curved sheet of paper. Surfaces matter because they help mathematicians and scientists describe and understand the properties of objects and spaces, including both everyday shapes and more abstract mathematical structures that don't necessarily exist in physical space.
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An open surface with x-, y-, and z-contours shown.
In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.