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

Every compact metric space is of finite topological dimension.

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.

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

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

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

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

In any closed and bounded subset of Euclidean space in dimension $n$, any continuous function mapping the set into itself must have a fixed point.

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

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