Totalraum

Totalraum refers to a concept in topology and geometry. It signifies a space where any point can be reached from any other point through a continuous path. More formally, a topological space is called a totalraum if it is connected and locally path-connected. A connected space is one that cannot be partitioned into two non-empty disjoint open sets. Locally path-connected means that every point has a neighborhood basis consisting of path-connected open sets.

The term is sometimes used interchangeably with "path-connected" or "connected" in certain contexts, but the precise definition emphasizes the combination of global connectivity and local path connectivity. This property ensures that the space is "well-behaved" in terms of paths and their continuity throughout the entire space. For instance, Euclidean spaces are totalraums.

In practical applications, understanding whether a space is a totalraum can be important in fields such as computer graphics, robotics, and theoretical physics when dealing with continuous motion or the structure of physical realities. It guarantees a certain degree of "stickiness" or continuity that might be essential for certain algorithms or theoretical models. The opposite of a totalraum would be a space that is disconnected or where certain regions are not accessible via continuous paths.