Überrounded

Überrounded is a concept in mathematics, specifically in the field of topology, that describes a type of space where every point is surrounded by a neighborhood that is homeomorphic to a closed ball. This concept is a generalization of the idea of a manifold, which is a space where every point has a neighborhood that is homeomorphic to an open ball. The term "überrounded" was coined by topologists to emphasize the difference between these two types of spaces.

In an überrounded space, the neighborhoods are closed, meaning they include their boundary points. This property distinguishes überrounded spaces from manifolds, where the neighborhoods are open and do not include their boundary points. The concept of überrounded spaces is particularly useful in the study of singularities and the behavior of functions on topological spaces.

One of the key characteristics of überrounded spaces is that they are locally compact, meaning that every point has a compact neighborhood. This property is crucial in many areas of mathematics, including the study of differential equations and the theory of measure and integration.

Überrounded spaces are also closely related to the concept of a locally Euclidean space, which is a space where every point has a neighborhood that is homeomorphic to a Euclidean space. The difference between überrounded spaces and locally Euclidean spaces lies in the nature of the neighborhoods: in überrounded spaces, the neighborhoods are closed, while in locally Euclidean spaces, they are open.

In summary, überrounded spaces are a type of topological space where every point is surrounded by a closed neighborhood that is homeomorphic to a closed ball. This concept is a generalization of manifolds and is useful in the study of singularities and the behavior of functions on topological spaces. Überrounded spaces are locally compact and are closely related to locally Euclidean spaces.