topological dimension

Topological dimension is a way to measure the complexity of a shape based on how it connects and covers space. Imagine a line (like a strand of string) has a dimension of one because you need one coordinate to specify a point on it. A surface (like a sheet of paper) has dimension two because two coordinates are needed. A solid object (like a cube) has dimension three. Topological dimension captures this idea in a broad sense, focusing on the number of parameters needed to describe local neighborhoods within a shape, regardless of size or curvature—it's about how many directions or degrees of freedom the shape inherently possesses.