What is topopolis?

Topopolis is a concept in geometry and topology that refers to the study of topological properties of certain sets of points in n-dimensional space. It was introduced in the mid-20th century by Austrian-born mathematician Karl Menger.

In topopolis, points in a space are replaced by neighborhoods of points, leading to a more abstract and generalized approach to topological concepts. Topopolis can be used to study the relationships between the sets of points and their neighborhoods, as well as the topological properties of the neighborhoods themselves.

One of the key features of topopolis is its emphasis on the connectivity and shape of sets of points, rather than on the precise location of individual points. This allows for a more flexible and versatile approach to understanding the topological properties of spaces.

Topopolis has applications in various fields of mathematics, including geometric topology, point-set topology, and knot theory. It is also used in computer science and architecture to study the spatial and topological properties of complex structures and networks.