Surface (topology)

The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres.

For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface.

In most writings on the subject, it is often assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second-countable, and Hausdorff.

More generally, a (topological) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H2 in C. These homeomorphisms are also known as (coordinate) charts.

The collection of such points is known as the boundary of the surface which is necessarily a one-manifold, that is, the union of closed curves.

The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally.

In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides".

In differential and algebraic geometry, extra structure is added upon the topology of the surface.

This added structure can be a smoothness structure (making it possible to define differentiable maps to and from the surface), a Riemannian metric (making it possible to define length and angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface—in which case the surface is called a Riemann surface), or an algebraic structure (making it possible to detect singularities, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology).

Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed extrinsic.

In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean.

A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space.

The image of a continuous, injective function from R2 to higher-dimensional Rn is said to be a parametric surface.

For example, in each polygon below, attaching the sides with matching labels (A with A, B with B), so that the arrows point in the same direction, yields the indicated surface.

The four models above, when traversed clockwise starting at the upper left, yield Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon (square).

The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces.

The connected sum of two surfaces M and N, denoted M # N, is obtained by removing a disk from each of them and gluing them along the boundary components that result.

In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism.

Topological and combinatorial proofs in general rely on the difficult result that every compact 2-manifold is homeomorphic to a simplicial complex, which is of interest in its own right.

The most common proof of the classification is (Seifert & Threlfall 1980),[1] which brings every triangulated surface to a standard form.

This was originally proven only for Riemann surfaces in the 1880s and 1900s by Felix Klein, Paul Koebe, and Henri Poincaré.

The precise locations of the holes are irrelevant, because the homeomorphism group acts k-transitively on any connected manifold of dimension at least 2.

Conversely, the boundary of a compact surface is a closed 1-manifold, and is therefore the disjoint union of a finite number of circles; filling these circles with disks (formally, taking the cone) yields a closed surface.

As a simple example, a non-compact surface can be obtained by puncturing (removing a finite set of points from) a closed manifold.

Perhaps the simplest example is the Cartesian product of the long line with the space of real numbers.

The Prüfer manifold may be thought of as the upper half plane together with one additional "tongue" Tx hanging down from it directly below the point (x,0), for each real x.

It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E2.

Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry.

A Riemannian metric endows a surface with notions of geodesic, distance, angle, and area.

For example, there are uncountably many non-isomorphic compact Riemann surfaces of genus 1 (the elliptic curves).

An open surface with x -, y -, and z -contours shown.
A sphere can be defined parametrically (by x = r sin θ cos φ , y = r sin θ sin φ , z = r cos θ ) or implicitly (by x 2 + y 2 + z 2 − r 2 = 0 .)
A knotted torus.
Some examples of orientable closed surfaces (left) and surfaces with boundary (right). Left: Some orientable closed surfaces are the surface of a sphere, the surface of a torus , and the surface of a cube. (The cube and the sphere are topologically equivalent to each other.) Right: Some surfaces with boundary are the disk surface , square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.