# Topology

**Topology** is a branch of advanced mathematics devoted to spatial relationships. It can also be described as the manipulation and mapping of sets. Traditionally the field was divided into the fields of continuous topology and geometric topology. Now, however, the subject of topology is divided into:

- algebraic topology
- point-set topology

Topology also has a specific mathematical definition as a collection of open sets in a topological space. The open sets can be manipulated, forming the basis of the topology. The opposite of open sets are closed sets. Topology may also be defined in terms of closed sets, but this yields an equivalent definition.

In topology, a *genus* of a surface is the greatest number of distinct, continuous closed curves that may be drawn on it without separating the surface into distinct regions. The closed curves cannot be self-intersecting. The genus of the surface of a sphere is 0, while the genus of a torus (doughnut shape) is 1.

If you draw a circle on a sphere, then you can put a point A inside the circle and a point B outside the circle. The only way to connect A to B would have to cross the circle. So drawing the circle divides the surface of the sphere into two regions. But you can draw a circle around the edge (rim) of a torus without creating two regions. You can get from point A on one side of the line to point B on the other side, by going through the hole in the center.

The field of topology was arguably founded by work done by Leonard Euler as published in 1736, based on a problem called “The Seven Bridges of Königsberg.”

## Introduction

Topology is a branch of mathematics. Like geometry, it studies shapes, but it focuses on certain properties of them. For example, length is not a topological property, so to a topologist, a line of length 20 metres and one of length 2 metres are the same. Nor are angles, or curvature, so a topologist would say that triangles and circles are also undistinguishable.

A way of thinking about topology (albeit a non-rigorous one) is that two shapes are topologically the same, if they can be transformed into one another if one can be turned into the other by stretching it (or parts of it), but there must be no cutting or gluing of the shape. If you imagine the objects are made of clay then you can stretch or compress the clay or pull it around, but you cannot tear it at any point.

## Applications

Applications of topology include:

- optimizing networks, such as internet traffic, telephonic communications, gene regulation and robotics
- managing data, such as surveys, or sensors
- analyzing dynamics, such as stochastic systems