Menger–Urysohn 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 Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.
We want the dimension of a point to be 0, and a point has empty boundary, so we start with
Then inductively, ind(X) is the smallest n such that, for every and every open set U containing x, there is an open V containing x, where the closure of V is a subset of U, such that the boundary of V has small inductive dimension less than or equal to n − 1. (In the case above, where X is Euclidean n-dimensional space, V will be chosen to be an n-dimensional ball centered at x.)
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