What Is Topology?

What is Topology?

Topology is a branch of geometry. Euclidean geometry studies properties of objects that are invariant under rigid motions in Euclidean space. Topology studies geometric properties that are invariant under any continuous deformation. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken. For example, a square can be deformed into a circle without breaking it, but a figure 8 cannot. Hence a square is topologically equivalent to a circle, but different from a figure 8.

A typical question in topology would be how many holes are there in an object. How can you define the holes in a torus or sphere? What is the boundary of an object? Is a space connected? Does every continuous function from the space to itself have a fixed point?

Topology is a relatively new branch of mathematics; most of the work has been done since 1900.

Types of Topology

General Topology or Point Set Topology

General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense.

Combinatorial Topology

Combinatorial topology considers the global property of spaces, built up from a network of vertices, edges and faces. This is the oldest branch of topology, and dates back to Euler. He showed that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic; this is the number (V - E + F), where V, E, and F are the number of vertices, edges and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any dissection of a sphere will have an Euler characteristic of 2.

Algebraic Topology

Algebraic topology also considers the global property of spaces, and uses algebraic objects, such as groups and rings, to solve geometric problems. Algebraic topology converts a topological problem into an algebraic problem, that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and Klein bottle can be distinguished from each other because they have different homology groups.


Torus	Klein Bottle

Algebraic topology often uses the combinatorial structure of a space to calculate the various groups associated to that space.

Differential Topology

Differential topology considers spaces with some kind of smoothness associated to each point. In this case, the square and the circle would not be differentiably equivalent to each other. Differential topology is useful for looking at vector fields, such as a magnetic or electrical fields.

Uses of Topology

Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex variables. It is also used in string theory in physics, and for describing the space-time structure of universe.

What is Pure Mathematics?