Dimension theory

Inductive dimension

In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets. The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to refer to the Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal. (Wikipedia).

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Related pages

Topological space | Metric space | Closure (topology) | Separable space | Topology | Second-countable space | Boundary (topology) | Hausdorff space | Sphere | Euclidean space | Georg Nöbeling | Mathematical induction | Closed set | Subset | Lebesgue covering dimension | Compact space | Irrational number | Normal space | Open set