Topology

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Topology

Topology is the study of properties of objects that do not depend on geometric measurements, and that do not change when the object is stretched or distorted without tearing. Topology is divided into three subdisciplines, which are largely different from each other: point-set topology, algebraic topology, and differential topology.

Point-set topology is the broadest and most fundamental of the different types of topology, and also the most axiomatic. The basic notion in point-set topology is the idea of continuity. The most classic definition of continuity is the one that arises in single-variable calculus, concerning functions that map the real numbers to themselves: a function is continuous, loosely speaking, if its graph has no jumps. This definition can be extended to a more general setting, for any function that maps one topological space to another. These more general functions are called continuous when they map one space to the other space without any tearing. Thus, a function that sends the western hemisphere of the earth onto a flat disk in way that we see on maps is continuous, since this mapping distorts the hemisphere but does not tear it. A function that flattens the entire globe onto a map on a sheet of paper is not continuous, since it must cut the sphere in order to flatten it.

To study continuity in its most general sense, mathematicians had to decide what kind of spaces permit the notion of continuous maps. The mathematical definition of 'not tearing', again speaking loosely, is that points that are close to each other in the first space should still be close to each other when they are mapped to the second space. Thus, in order for a space to be able to have continuous maps, the space must come equipped with a notion of closeness. It is hard to imagine a space that does not come with a natural definition of closeness, but when a space is defined in an abstract way, instead of by a concrete picture, there is not always one definition of closeness that is more natural than others. The study of spaces that come equipped with a notion of closeness is point-set topology. In this discipline, mathematicians use an axiomatic approach--they try to make as few assumptions as possible about the nature of the spaces they study (in particular, they try not to rely on their intuition, which is often misleading), and prove theorems that are as strong as possible. In this stripped-to-the-basics approach, even obvious-sounding statements must be proved from scratch, and their proofs can be surprisingly difficult. For example, one of the main theorems of point-set topology is the Jordan curve theorem, which states that a closed loop in the plane always divides the plane into an inside and an outside--this sounds straightforward, but it is not, since it is a statement about every possible loop, no matter how complicated.

Algebraic topology, which uses algebraic techniques to distinguish one topological space from another, is perhaps the oldest branch of topology. The first work that can be considered to be topology is a 1736 paper of Leonhard Euler, in which he solved the problem of the Königsberg bridges, a question about whether it is possible to walk through the city of Königsberg in such a way that each of its seven bridges is crossed once and only once. This is a question of topology, not geometry, since it is more important how the bridges are connected to each other than their precise geometric measurements. Euler proved that it is impossible to make such a walk, using algebraic methods: he showed that in a city where such a walk is possible, each island must have an even number of bridges coming out of it, except the islands on which the walk started and ended; this was not the case in Königsberg, so the walk was impossible.

In 1750, Euler published an even more influential paper in which he studied polyhedra from an algebraic point of view. In a convex polyhedron (a polyhedron for which any line connecting two points of the polyhedron is contained in the polyhedron), Euler discovered the relationship v-e+f=2, where v is the number of vertices (endpoints), e is the number of edges, and f is the number of faces (sides). In 1813, Antoine-Jean L'Huilier realized that Euler's formula was wrong for polyhedra that have holes in them, such as polyhedra that are shaped roughly like the surface of a donut. L'Huilier discovered that if the number of holes (or tunnels) in the polyhedron is denoted g, then v-e+f=2-2g. This was the first topological invariant-a number that could be attached to a topological space, that only depended on its topology, not its geometry. This number, called the Euler characteristic of the surface, gave a way to distinguish between two topological surfaces: if they had different Euler characteristics, then they must be different spaces. This invariant was the beginning of a branch of algebraic topology called homology theory, which tries to distinguish between spaces by examining polyhedra (perhaps higher-dimensional) that can be placed on (or in) the space, and by using them to study what kind of 'holes' the space has. Another similar technique is called homotopy theory: it uses circles, spheres, and higher-dimensional spheres to measure how many holes a given space has.

The third branch of topology is differential topology, which applies the notions of calculus to topological spaces. Differential topology is studied on a more restricted class of spaces than are point-set or algebraic topology: spaces called 'smooth manifolds'. A smooth manifold is a space for which at every point, there is a small neighborhood of that point that can be mapped to a small disk in Euclidean space, and, when two of those neighborhoods intersect, the corresponding maps of Euclidean space are smooth (differentiable). Although this is a substantial restriction on the kinds of spaces that are studied, the power of the techniques of differential topology make this a worthwhile restriction. Many of the topological spaces that are most important to physics are smooth manifolds, and can be understood using these techniques.