Dimension

A topological dimension is a variable required in order to specify points' locations within a space.

When a space has n dimensions, it can be categorized as an n-dimensional space. For example, the surface of a sphere is 2 dimensional; longitude and latitude are needed to specify where a point is on a sphere, but a line only requires one variable. In general, the dimensionality of a space is the minimum number of coordinates required to uniquely define any point inside it.

Dimension in Polytopes
"Main article: Polytope"The dimensionality or rank of a polytope is defined as the largest integer greater than the dimensionality of said polytope's elements. It is usually assumed that the point is 0-dimensional or that every polytope has the nullitope, which has dimension -1. For the most part, a n-dimensional polytope can be contained within a n-dimensional topological space.