topólogía

Topología is a branch of mathematics concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, and bending, but not tearing or gluing. These deformations are called homeomorphisms. A homeomorphism is a function between two topological spaces that is continuous, has a continuous inverse, and is bijective. The study of topology can be seen as a generalization of geometry where distances and angles are not as important as connectivity and the presence or absence of holes.

The fundamental objects of study in topology are topological spaces, which are sets equipped with a topology. A topology on a set is a collection of subsets, called open sets, that satisfy certain axioms. These axioms include that the empty set and the set itself must be open, the union of any number of open sets must be open, and the intersection of any finite number of open sets must be open. Properties that are preserved under homeomorphisms are called topological invariants. Examples of topological invariants include the number of connected components, the Euler characteristic, and the fundamental group.

Topology has applications in various fields, including physics, computer science, and data analysis. In physics, it is used in areas like string theory and condensed matter physics. In computer science, topological sorting is an algorithm used to order the nodes of a directed acyclic graph. In data analysis, topological data analysis (TDA) uses topological concepts to understand the shape and structure of data.