General topology is the branch of mathematics which formalizes and explores the ideas of approximity and limit. For this reason, it is a basic tool for the development of many other branches of mathematics.

Its origin lies in Cantor papers published between 1879 and 1884 concerning with problems of uniqueness for trigonometric series. At the beginning of the 20th century, Fréchet and Riesz introduced independently, the concept of abstract space endowed with a topological structure. However the first satisfactory definition about topological spaces using the concept of neighbourhoods was given by Hausdorfff in 1914.

Until the middle of the 20th century, metric compact­ spaces and uniform spaces attracted the attention of the researchers in this area. In 1962, Pervin proved that every topological space is quasiuniformable. This important result increased significantly the interest in asymmetric topological structures (quasiuniformities, quasimetrics, quasinorms, etc.). In recent years, these structures have had interesting applications in algorithmic complexity, computational programming, computational biology, etc.

IUMPA has a group of researchers dedicated to topology and its applications with two existing competitive projects (MEC, coordinated with a group of the “Universidad del País Vasco” and other group funded by “Generalitat Valenciana”). The main research topics are: asymmetric structures, the hyperspaces, metrics and fuzzy quasimetrics, as well as connections to algebraic structures and applications to various fields of the computer science (complexity of algorithms and programs, semantics, databases, etc.).

 

Research Groups: Topology and its Applications