Topology is a branch of mathematics. Its purpose is to study the structure of objects without paying attention to their size and initial shape, just as geometry does. Geometry mathematically describes a figure and topology analyzes the possibilities of figures. Let's think about a circumference. On the one hand, it is a figure in which all the points are at the same distance from the center. If the circumference were in three dimensions and were a ball it could be turned into a cube.
Topology understands objects as if they were made of rubber and could be transformed. In fact, the properties of objects remain unchanged even though their shape is alterable. If we think of a circle, it is a geometric figure but if we can manipulate it it becomes another figure: a triangle or an ellipse. This concrete example gives a guide to a basic principle of topology: equivalence between figures. Two figures are equivalent if one is convertible into another.
If we start from the idea that the surfaces of objects are modifiable (think of a sheet of paper that can be cut or folded), it is easy to see that the specific applications of topology are immense. In computing, programs are used to modify images. In optics the structure of the lenses is altered. In industry, objects are subject to variations in their shapes.
These examples demonstrate the versatility of the topology.
From a theoretical point of view, topology is related to other mathematical operations (statistics, differential equations ...). However, what is striking about topology is its ability to solve practical problems: analyze the best route for the delivery of goods or how to modify an object without breaking it. At the same time, topology has provided a very useful model and basic structure for biology, specifically for the explanation of DNA. The genetic material is distributed in two complementary chains, the double helix, which are wound through the same axis. And the curvature of the axis is a topological shape.
In conclusion, topology is based on a series of theoretical and abstract principles and from these it is possible to apply them to a multitude of areas of knowledge. In fact, despite the complexity of this branch of mathematics, according to psychology, children intuitively handle the principles of topology in their games and in the manipulation of objects.
