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A ratio providing a statistical index of complexity comparing how the detail in a pattern changes with the scale at which it is measured.

A fractal with Hausdorff dimension that is a non-integer, which is typically greater than its topological dimension.

Adding a point which does not lie on the space can increase its Lebesgue covering dimension by one.

Defined using the concept of measure at different scales, describing a fractal's 'roughness' or 'fragmentation'.

The minimum value of n such that any open cover has a refinement where no point is included in more than n+1 sets.

Equivalent to the small inductive dimension and defined using the concept of local separation on a topological space.

The number of local coordinates necessary to specify points near any point in the space.

Defined recursively using the separation properties of the boundary of a small open set around a point.

A minimal set of points in a metric space such that all other points can be uniquely determined by their distances to the points in this set.

An example of a set with topological dimension zero but an uncountably infinite number of points.

Every compact metric space is of finite topological dimension.