There are various properties of a figure, in general, and of a surface such as a sphere, torus, or disk, in particular, that may be used to distinguish between such figures topologically. One property is the number of boundaries the surface has, if any. Another property is orientability; a surface is orientable if a circle drawn on it with a given orientation (clockwise or counterclockwise) always, if moved around the surface, returns to its original position with the same orientation. A sphere and a torus are both orientable, but a Möbius strip (a one-sided surface made by twisting a strip of paper and joining the ends so that opposite edges correspond) is a nonorientable surface, since an oriented circle moved around the strip will return to its original position with its orientation reversed (see Möbius, Augustus Ferdinand).
Another topological property of a surface is its Euler-Poincaré characteristic, a number which can be calculated from any polyhedral decomposition of the surface. If V is the number of points (vertices) in the decomposition, E is the number of line segments (edges), and F is the number of regions (faces), then the characteristic is given by χ = V - E + F and is the same for all possible polyhedral decomposition of the given surface. For a sphere, χ = 2, and the formula is identical with Euler's formula for the vertices, edges, and faces of a spherical polyhedron, to which the sphere is topologically equivalent. For a torus, χ = 0. The Euler-Poincaré characteristic for an orientable surface is χ = 2 - 2 p, where p is called the genus of the surface. Any orientable closed surface is topologically equivalent to a sphere with p handles attached to it; e.g., the torus, having χ = 0, is of genus 1 and is equivalent to a sphere with one handle, and a double torus (two-hole doughnut), equivalent to a sphere with two handles, is of genus 2 and has χ = - 2. For a nonorientable surface, χ = 2 - q, where q is the number of cross-caps that must be added to a sphere to make it equivalent to the surface. (A cross-cap is a cap with a twist like a Möbius strip in it.)
Closely related to the Euler-Poincaré characteristic is the connectivity number of a surface, which is equal to the largest number of closed cuts (or cuts connecting points on boundaries or on previous cuts) that can be made on the surface without separating it into two or more parts. The connectivity number is equal to 3 - χ for a closed surface and to 2 - χ for a surface with boundaries (e.g., a disk). A surface with a connectivity number of 1, 2, or 3 is said to be simply connected, doubly connected, or triply connected, respectively, and similarly for more complex surfaces; a sphere is simply connected, while a torus is triply connected. Thus, any surface can be classified by its boundary curves (if any), its orientability, and its Euler-Poincaré characteristic or connectivity number; and any surface is topologically equivalent to a sphere with an appropriate number of handles, cross-caps, or holes. A surface is a simple example of a topological space, the basic entity studied in topology.
Different types of topological spaces are defined according to axioms satisfied by the sets of points that constitute the space. Especially important are topological spaces for which a distance function is defined for every pair of points in the space; such spaces are called metric spaces. A full treatment of the properties of topological spaces of arbitrary dimension requires various concepts of an advanced nature, e.g., homology theory, and is beyond the scope of a general article. The most important spaces, manifolds, are those which are locally equivalent to the Euclidean space of the same dimension. The fundamental problem of classifying manifolds was classically solved for dimensions 1 and 2, and largely clarified in dimensions 5 or more during the past 30 years. Dimensions 3 and 4 are now areas of vigorous research, stimulated in part by ideas from physics. The theory of knots plays an important role in dimension 3, and has revealed surprising connections with physics and application to biology.
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