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Topological Dimension Theory
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Topological Dimension
The number of local coordinates necessary to specify points near any point in the space.
Lebesgue Covering Dimension
The minimum value of n such that any open cover has a refinement where no point is included in more than n+1 sets.
Inductive Dimension
Defined recursively using the separation properties of the boundary of a small open set around a point.
Hausdorff Dimension
Defined using the concept of measure at different scales, describing a fractal's 'roughness' or 'fragmentation'.
Fractal Dimension
A ratio providing a statistical index of complexity comparing how the detail in a pattern changes with the scale at which it is measured.
Alexandroff's Theorem
Every compact metric space is of finite topological dimension.
Menger-Urysohn Dimension
Equivalent to the small inductive dimension and defined using the concept of local separation on a topological space.
Dimension Raising
Adding a point which does not lie on the space can increase its Lebesgue covering dimension by one.
Cantor Set
An example of a set with topological dimension zero but an uncountably infinite number of points.
Sierpinski Triangle
A fractal with Hausdorff dimension that is a non-integer, which is typically greater than its topological dimension.
Brouwer's Fixed Point Theorem
In any closed and bounded subset of Euclidean space in dimension , any continuous function mapping the set into itself must have a fixed point.
Metric Dimension
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.
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