For questions about topological or metric notions of dimension, including the Lebesgue, small and large inductive, Hausdorff, and packing dimensions, etc. and for questions regarding fractal geometry and analysis. Use [linear-algebra] instead for questions about the dimension of vector spaces. Also note [dimension-theory-algebra].
Dimension theory studies various notions of dimension defined for topological (and, more specifically, metric) spaces. These notions of dimension include the Lebesgue covering dimension, the small and large inductive dimensions, and the Hausdorff dimension, among others.
Broadly speaking, these notions of dimension are invariants of topological or metric spaces which seek to quantify the relation between the diameter of a set in a metric space, and the volume or measure of that set. For example, a ball in $\mathbb{R}^3$ is three-dimensional, as the volume of the ball is proportional to the cube of the radius—that is, $\operatorname{vol}(B_r) \propto r^3$. The dimension "sees" the cubic scaling law. Other notions of dimension generalize this basic idea to spaces where the volume may not be a priori defined, or where the scaling law involves non-integer powers.
Dimension theory is often related to the study of fractals, hence this tag might be appropriate for questions about fractals, fractal geometry, and analysis on fractals. This tag is not intended for questions about the dimension of a vector space, which are better tagged with linear-algebra. There is also the dimension-theory-algebra for questions about algebraic notions of dimension, such as Krull dimension.
Related tags: hausdorff-dimension, fractals
